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# Physics and Astronomy Labs/Uniform acceleration tapping basketball with spitwads

Student Hitting a Ball[1]

In the Tapping Basketballs and Spitwad Lab we started by marking every 5 tiles in the hallway. We designated people to different jobs. One person tapped the back of a basketball at a constant rhythm with a meter stick so the basketball would accelerate at a constant pace. Other people made spitwads. One person was a timer we used the spitwads to mark where the ball was at a certain time. we used seconds, so, every second the timer yelled and the throwers threw spitwads to mark where the ball was. After everyone threw we marked the distances of the best spitwads from the origin we then used the data to make a scatterplot and then found the line of best fit.

In the lab, we had someone hit a basketball with a meter stick which caused it to accelerate, while this was happening, we had two people on each side of the hall assigned with a number throw spit wads in front of the ball as their number was called.

When everybody's number was called and the spits wads were thrown, we measured the distance between each second using the spit wads on the ground, this helped us figure out the uniform acceleration of the ball.

## Data taken by students in How Things Work

Can you estimate the slope and guestimate the uncertainty?
 interval t (sec) v(ft/s) ft in 0 0 '0-1 0.5 2 1 '1-2 1.5 6 10 '2-3 2.5 9 6 '3-4 3.5 8 6 '4-5 4.5 14 7 '5-6 5.5 10 5

## Trig Physics

To be modified pending input from Calc Physics

### GW

We took data from an experiment, which consisted of using a ruler to hit a ball--marking where the ball has travelled with spit wads thrown every second, and attempting to increase velocity at a steady rate. The distance between each spit wad is the distance the ball has travelled in one second. The data included time and distance. We then put that data into an excel spreadsheet, solved for the approximate velocity of the ball, then made a velocity-time graph to map out the experiment. Finally, we solved for two different rates of change to better capture the entire span of the graph, then solved for our percent error.

### TJ

Data were collected from an experiment conducted by using a yard stick to hit a basketball down the hall while spit wads were used to measure the distance in feet. Data from the lab was inserted into excel and the velocity was calculated in ft/s. Once a graph was constructed two lines of best fit were drawn and the slope of both were found using the change in velocity divided by the change in time. using this information we guesstimated the uncertainty.

### TR

1. Copied data from this table into Microsoft Excel
2. Inserted formula into velocity column to calculate velocity; =(ft+in/12)/(1)
• Divided by 1 because the time interval between measurements was 1 sec. 3.

## Calculus physics

1. Suggest an edit on the lab as described above.
2. Measure the slope with an uncertainty in the figure by hand. Keep in mind that "uncertainty" usually means one standard deviation, suggesting that typically 68% of the data points fall within your range.
Repeat using Excel and/or Matlab
3. Explain the formulas governing this. Use pencil and paper on equations and sketches. Put words into sandbox, referencing the equations and sketches.

### finding the uncertainty in slope

Method 1; entire class measures slope by hand and we take the mean and standard deviation
 student 1 2 3 4 5 6 7 8 9 10 11 slope(m/sQ) 2.80 2.67 3.00 3.33 3.00 2.67 2.80 3.00 3.20 2.67 3.67 average stdev 3.0 0.3
Metho 2; Draw a "high" and "low" estimate.

Result: ${\displaystyle 3.0\pm 0.2}$

## How Things Work Lake Campus Fall 17

 beat 0 1 2 3 4 5 6 feet 0 5.2 10.2 17 27 40.6

Is this good data? Look at the 1 and 2 are the first (or among the first) HR diagrams to be published. This is a modern one.

This was graphed in two courses and the acceleration was "eyeball fit" to about 2.0±0.5 ft/sec2. We had to "cheat" by including a zero velocity data point at t = 0 and also neglected the last data point. Note: This is not "cheating" if you report what you did, and also, no scholarly paper would use the word "cheat" to explain what was done here.--user:Guy vandegrift

## At many small strikes equivalent to a gentle push (or force)?

bla
bla bla.

This "permalink" cannot be edited and the section linked to has been approved by user:Guy vandegrift: w:special:permalink/798712661#Kinetic_theory_of_gases

### Footnotes

1. If this photo looks staged, it is. They already took data before we could take a picture.


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# 1.4: Welfare Economics - Consumer and Producer Surplus

## Introduction to Elasticities

An elasticity is a measure of responsiveness.

Elasticity = How responsive one variable is to a change in another variable.

An elasticity $$(E)$$, or responsiveness, is measured by the percentage change of each variable. The change in a variable is the ending value $$(X_1)$$ minus the initial value $$(X_0)$$, or $$ΔX = X_1 – X_0$$. A percentage change in a variable is defined as the change in the variable divided by the initial value of the variable:

$\%ΔX = \dfrac{ΔX}{X_0}.$

Using this formula, Equation \ref{1.7} shows the responsiveness of $$Y$$ to a change in $$X$$:

$E = \frac{\%ΔY}{\%ΔX} = \frac{(ΔY/Y)}{(ΔX/X)} = \left(\frac{ΔY}{ΔX}\right)\left(\frac{X}{Y}\right). \label{1.7}$

Elasticities can be calculated for any two variables. Elasticities are widely used in economics to measure how responsive producers and consumers are to changes in prices, income, and other economic variables. Elasticities have a very desirable property: they do not have units. Since the two variables are measured in percentage changes, the units of each variable are cancelled, and the resulting elasticity has no units. This allows elasticities to be compared to each other, when prices and quantities cannot be directly compared. For example, the quantity of apples cannot be directly compared to the quantity of orange juice, since they are in different units. However, the elasticities of oranges and apples can be compared directly, since there are no units for elasticities.

### Own Price Elasticity of Demand: Ed

The own-price elasticity of demand (most often called simply the “price elasticity of demand” or the “elasticity of demand”) measures the responsiveness of consumers to a change in price, as shown in Equation \ref{1.8}:

$E_d = \frac{\%ΔQ^d}{\%ΔP} = \left(\frac{ΔQ^d}{ΔP}\right)\left(\frac{P}{Q^d}\right).\label{1.8}$

Own Price Elasticity of Demand = the percentage change in quantity demanded given a one percent change in the good’s own price, ceteris paribus.

The own-price elasticity of demand is the most important thing that a business firm can know. The price elasticity informs the business about how a change in price will affect the quantity demanded. If consumers are responsive to price changes, the firm may think twice before raising the price and losing customers to the competition. On the other hand, if consumers are relatively unresponsive to price changes, the firm may increase the price, and most customers will continue to purchase the good at the higher price. Food is an example of an inelastic good, since we all need to eat.

The price elasticity of demand $$(E_d)$$ depends on the availability of substitutes. If there are no substitutes for a good (food, toilet paper, toothpaste), the good is called, “price inelastic.” Consumers will purchase the good even at a high price. If substitutes are available, the good is considered to be “price elastic:” a higher price will cause customers to decrease consumption of the good by buying the substitute good. Green shirts are an example: if the price of green shirts is increased, consumers will shift purchases to blue shirts, or shirts of a different color.

The price elasticity of demand is the most critical aspect of a business firm, since it provides the most crucial information about customers! Knowledge of the price elasticity of demand provides information to a business firm on how consumers would react to price changes, allowing the firm to identify the profit-maximizing price to charge consumers.

### Price Elasticity of Demand Example

Suppose that the price of wheat is equal to USD 4/bu of wheat, and increases to USD 6/bu. Due to the higher price, suppose that wheat millers reduce their purchases of wheat from 10 million bushels (m bu) to 8 million bushels. The price elasticity of demand for wheat can be calculated using Equation \ref{1.9}. By convention, the initial values of $$P$$ and $$Q^d$$ are used in the elasticity calculation for the variables $$P$$ and $$Q^d$$.

$E_d = \frac{\%ΔQ^d}{\%ΔP} = \frac{(ΔQ^d/ΔP)}{(P/Q^d)} = \left(\frac{\text{8-10 m bu}}{\text{6-4 USD/mt}}\right)\cdot\left(\frac{\text{4 USD/bu}}{\text{10 m bu}}\right)\label{1.9}$

Notice that the units cancel: there are (m bu) in both the numerator and denominator, and (USD/bu) also appears in both numerator and denominator. This allows the math to be greatly simplified:

$E_d = \left(\frac{-2}{2}\right)\cdot\left(\frac{4}{10}\right) = \frac{-1}{0.4} = -0.4$

The price elasticity of demand is always negative, due to the Law of Demand. By convention, economists take the absolute value to make $$E_d$$ positive. For example, in this case, $$E_d= – 0.4$$, then $$\mid E_d \mid = 0.4$$. The own price elasticity of demand provides important information about the wheat market: how responsive wheat buyers are to a change in price. To interpret the elasticity, it means that for a one percent increase in price, the quantity demanded of wheat will decrease by 0.4 percent. This is a relatively inelastic response, since the change in quantity demanded is smaller than the price change.

Elasticities are classified into three categories, based on consumer responsiveness to a one percent change in price.

\begin{align*} &\text{Price elastic } & \mid E_d \mid &> 1\\ &\text{Price inelastic} & \mid E_d \mid &< 1\\ &\text{Unitary elastic} & \mid E_d \mid &= 1\end{align*}

Goods that are price elastic have substitutes available, and the percentage change in quantity demanded will decrease more than the percentage increase change in price $$(\%ΔQ^d > \%ΔP$$, therefore $$\mid E_d\mid > 1)$$. A price inelastic good, on the other hand, will have a smaller percentage change in quantity demanded than the percentage increase in price $$(\%ΔQ^d < \%ΔP$$, therefore $$\mid E_d \mid < 1)$$. For unitary elastic goods, the percentage change in quantity demanded is equal to the percentage change in price $$(\%ΔQ^d = \%ΔP$$, therefore $$\mid E_d\mid = 1)$$.

### Elastic and Inelastic Demand Examples

To compare elastic and inelastic demands, think of a student who would like to purchase a pack of cigarettes during a late night study session for an exam. If the student arrives at the convenience store to find that the price of Marlboros, her usual brand, has doubled, she could switch to many other brands: Lucky Strikes, Winstons, etc. The demand for Marlboro cigarettes is price elastic (left panel, Figure $$\PageIndex{1}$$). The price elasticity of demand depends on the availability of substitutes. An elastic demand will have a relatively flat slope, since a small change in price results in a relatively larger change in quantity demanded.

On the other hand, if the convenience store increases the price of all cigarettes, the student will pay for a pack, since there are no substitutes for all cigarettes (right panel, Figure $$\PageIndex{1}$$). More narrowly defined goods will have larger absolute values of own price elasticities, since there are more substitutes for narrowly defined goods. For example, apples are more price elastic than all fruit, and green shirts are more price elastic than all shirts. An inelastic good will have a steep slope, since the change in quantity demanded is small relative to the change in price.

Figure $$\PageIndex{2}$$ shows a range of own price elasticities, from perfectly inelastic to perfectly elastic.

A good that is perfectly inelastic is one that consumers purchase no matter what the price is. Within a certain range of prices, this could be food or electricity. In this case, quantity demanded is completely unresponsive to changes in price: $$\mid E_d\mid = 0$$. An inelastic demand is one where the percentage change in price is larger than the percentage change in quantity demanded: $$\%ΔQ^d < \%ΔP$$, and $$\mid E_d\mid < 1$$. Goods that are price inelastic are characterized by consumers being unresponsive to price changes. Goods that are price elastic exhibit relatively high levels of consumer responsiveness to price movements. For elastic goods, the percentage change in quantity demanded is larger than the percentage change in price: $$\%ΔQ^d > \%ΔP$$, and $$\mid E_d\mid > 1$$. A perfectly elastic good is characterized by a horizontal demand curve. In this case, if the price of the good is increased even one cent, all customers decrease purchases of the good to zero. An individual wheat farmer’s crop is an example. If the farmer tries to raise the price by one cent more than the prevailing market price, no consumers would purchase her wheat. There are a large number of perfect substitutes available from other wheat farmers, so the price elasticity is infinite, and the good is called, “perfectly elastic.”

## Own Price Elasticity of Supply: Es

Producer responsiveness to a change in price is measured with the own price elasticity of supply, often called the price elasticity of supply, or the elasticity of supply (Es). The formula for the price elasticity of supply is given in Equation \ref{1.10}:

$E_s = \frac{\%ΔQ^s}{\%ΔP}. \label{1.10}$

Own Price Elasticity of Supply = the percentage change in quantity supplied given a one percent change in the good’s own price, ceteris paribus.

The own price elasticity of supply is always positive, because of the Law of Supply: there is a direct, positive relationship between the quantity supplied of a good and the good’s own price, ceteris paribus. Similar to the price elasticity of demand, the price elasticity of supply is categorized into three elasticity classifications.

\begin{align*} &\text{Price elastic } & E_s &> 1\\ &\text{Price inelastic} & E_s &< 1\\ &\text{Unitary elastic} & E_s &= 1\end{align*}

A good with an elastic supply is one where the percentage change in quantity supplied is greater than the percentage change in price: $$\%ΔQ^s > \%ΔP$$, and $$E_s > 1$$. Since $$E_s$$ is always positive, the absolute value is not necessary (redundant). A good with an inelastic supply has a smaller percentage change in quantity supplied, given a percent change in price: $$\%ΔQ^s < %ΔP$$, and $$E_s < 1$$. A good with unitary elasticity of supply has equal percent changes in quantity supplied and price: $$\%ΔQ^s = \%ΔP$$, and $$E_s = 1$$. Figure $$\PageIndex{3}$$ illustrates the different categories of the own price elasticity of supply.

## Income Elasticity: Ei

The income elasticity $$(E_i)$$ measures how consumers of a good respond to a one percent increase in income $$(I)$$, as shown in Equation /ref{1.11}:

$E_i = \frac{\%ΔQ^d}{\%ΔI}.\label{1.11}$

The income elasticity is defined in a similar way as the price elasticities.

Income Elasticity = the percentage change in demand given a one percent change in income, ceteris paribus.

Income elasticities are also categorized into responsiveness classifications. A normal good is one that increases with an increase in income $$(E_i > 0)$$. There are two subcategories of normal goods: necessities and luxury goods. Notice that necessity goods and luxury goods are normal goods. They represent subgroups of the normal category, since $$E_i$$ is positive in both cases.

\begin{align*} &\text{Normal Good } & E_i &> 0\\ &\text{Necessity Good} & 0 &< E_i < 1\\ &\text{Luxury Good} & E_i &> 1\\ &\text{Inferior Good} & E_i &< 1\end{align*}

The graphs of the relationship between income and demand are called, “Engel Curves,” named for Ernst Engel (1821-1896), a German statistician who first investigated the impact of income on consumption.

A necessity good is a normal good that has a positive, but small, increase in demand given a one percent increase in income. Food is an example, since consumers increase the consumption of food with an increase in income, but the total amount of food consumed reaches an upper limit. This is shown in Figure $$\PageIndex{4}$$, left panel.

A luxury good is one that has increasing demand as income increases, as shown in the right panel of Figure $$\PageIndex{4}$$. Good such as boats, golf club memberships, and expensive clothing are examples of luxury goods. Inferior goods (Figure $$\PageIndex{5}$$) are characterized by lower levels of consumption as income increases: ramen noodles and used clothes are examples.

It is important to point out that a good can be a normal good at low income levels, and an inferior good at higher income levels. Hamburger (ground beef) is an example. At low levels of income, hamburger consumption might increase when income rises (Figure $$\PageIndex{6}$$). However, at higher levels of income, consumers might shift out of ground beef and into more expensive meats such as steak. Figure $$\PageIndex{6}$$ shows that the same good can be both a normal good and an inferior good, at different levels of income.

## Cross Price Elasticity of Demand: Edxy

The cross price elasticity of demand measures the responsiveness of demand for one good with respect to a change in the price of another good.

$E_{dxy} = \frac{\%ΔQ^d_y}{\%ΔP_x}\label{1.12}$

Cross Price Elasticity of Demand = the percentage change in the demand of one good given a one percent change in a related good’s price, ceteris paribus.

The cross price elasticity is important for two categories of related goods: substitutes and complements in consumption. Substitutes in consumption will have a positive cross price elasticity of demand, since consumers will decrease purchases of the good that has the price increase, and buy more substitute goods. Complements in consumption are goods that are consumed together, like macaroni and cheese. If the price of macaroni increases, then consumption of both macaroni and cheese decreases.

\begin{align*} &\text{Substitutes in Consumption } & E_{dxy} &= \frac{\%ΔQ^d_y}{\%ΔP_x} > 0\\ &\text{Complements in Consumption } & E_{dxy} &= \frac{\%ΔQ^d_y}{\%ΔP_x} < 0\\ &\text{Unrelated Goods in Consumption } & E_{dxy} &= \frac{\%ΔQ^d_y}{\%ΔP_x} = 0 \end{align*}

Unrelated goods have a cross price elasticity of demand equal to zero. This is because a change in the price of a good has no effect on the quantity demanded of an unrelated good.

## Cross Price Elasticity of Supply: Esxy

The cross price elasticity of supply captures the responsiveness of the supply of one good, given a change in the price of another good.

$E_{sxy} = \frac{\%ΔQ^s_y}{\%ΔP_x} \label{1.13}$

Cross Price Elasticity of Supply = the percentage change in the supply of one good given a one percent change in a related good’s price, ceteris paribus.

Substitutes in production are goods that are produced “either/or,” such as corn and soybeans. The same resources (land, machinery, labor, etc.) could be used to produce either corn or soybeans, but the two crops can not be grown on the same land at the same time. The cross price elasticity of supply of substitutes in production is negative. If the price of corn increases, for example, then producers will devote more land to corn and less to soybeans.

\begin{align*} &\text{Substitutes in Production } & E_{sxy} &= \frac{\%ΔQ^s_y}{\%ΔP_x} < 0\\ &\text{Complements in Production } & E_{sxy} &= \frac{\%ΔQ^s_y}{\%ΔP_x} > 0\\ &\text{Unrelated Goods in Production } & E_{sxy} &= \frac{\%ΔQ^s_y}{\%ΔP_x} = 0 \end{align*}

Complements in production are goods that are produced together, such as beef and leather. Complements in production have a positive cross price elasticity: if the price of beef increases, both more beef and more leather will be supplied to the market. Unrelated goods in production have a cross price elasticity of supply equal to zero, since the price of an unrelated good has no impact on the demand of the other unrelated good.

## Price Elasticities and Time

The magnitude of the price elasticity of supply measures how easy it is for the firm to adjust to price changes. In the immediate run (a short time period), the firm can not adjust the production process, so the supply is typically perfectly inelastic. In the short run, a time period when some inputs are fixed and some inputs are variable, the firm may be able to adjust some inputs, so supply is inelastic, but not perfectly inelastic. In the long run, all inputs are variable, and the firm can make adjustments to the production process. In this case, supply is elastic. As more time passes, the price elasticity of supply increases.

This relationship also holds for the price elasticity of demand. If the price of a good increases in the immediate run, there is little consumers can do other than purchase the good. Air travelers who have an emergency that they need to attend to will pay a high price of an airline ticket on the same day of the flight. As time passes, there are more options available to the consumer, and the price elasticity of demand becomes more elastic with the passage of time.

## Elasticity of Demand along a Linear Demand Curve

Interestingly, the elasticity of demand changes along a linear demand curve. This is due to the calculation of the own price elasticity of demand as percentage change in quantity demanded caused by a percentage change in the price of the good. In Figure $$\PageIndex{7}$$, the slope of the demand function is constant: it does not change over the entire demand curve.

For example, suppose that the inverse demand function is given by: $$P = 10 – Q^d$$, where $$P$$ is the price of the good and $$Q^d$$ is the quantity demanded. In this case, the vertical intercept (y-intercept) is equal to 10, and the slope is equal to negative one. It should be emphasized that, in this case, the slope is constant and equal to minus one for the entire demand curve.

The elasticity of demand, however, changes in value quite dramatically from the y-intercept to the x-intercept. It changes from a value of zero on the x-axis to a value of negative infinity on the y-axis. The cause is the calculation of percentage change.

Consider an example of a mouse and an elephant. If both gain one pound, the weight gain is identical, but the percentage change is vastly different. Suppose that the mouse weighs one-tenth of a pound, the elephant weighs 10,000 pounds, and the total weight gain for both the mouse and the elephant is one pound. The percentage weight gain is $$\%ΔWG = \dfrac{ΔWG}{WG_0}$$, where $$ΔWG$$ is the change in weight, and $$WG_0$$ is the initial weight gain. For the mouse,

$\%ΔWG \text{ mouse } = \frac{ΔWG}{WG_0} = \frac{1 \text{ lb}}{0.1 \text{ lbs}} = 10 = 1000 \text{ percent!} \label{1.14}$

For the elephant,

$\%ΔWG \text{ elephant} = \frac{ΔWG}{WG_0} = \frac{1 \text{ lb}}{10,000 \text{ lbs}} = 0.0001 = 0.01 \text{ percent!} \label{1.15}$

The take home message of the story is that the total weight gain was identical for both the elephant and the mouse (one pound), whereas the percentage weight gain was enormously different.

This is also true of the elasticity of demand along the linear demand curve. Consider the point where the linear demand curve crosses the x-axis. At this point, the price is equal to zero. Suppose that we raised the price by one unit to find out how responsive consumers are to an increase in price. The price elasticity of demand is:

$E_d = \frac{\%ΔQ^d}{\%ΔP}.\label{1.16}$

At the x-intercept, the percentage change in price $$(\%ΔP)$$ is equal to $$dfrac{ΔP}{P} = \dfrac{1}{0} =$$ infinity. The elasticity of demand is equal to the percentage change in quantity demanded $$(\%ΔQ)$$ divided by the percentage change in price $$(\%ΔP =$$ infinity). Thus, $$E_d = 0$$ at the x-intercept, since dividing any number by infinity is equal to zero.

How responsive are consumers to a change in price at the vertical axis? At the y-intercept, the percentage change in quantity demanded $$(\%ΔQ_d)$$ is equal to $$\dfrac{ΔQ_d}{Q_d} = \dfrac{1}{0} =$$ infinity. Therefore, the elasticity of demand is equal to the percentage change in quantity demanded $$(\%ΔQ =$$ infinity) divided by the percentage change in price $$(\%ΔP)$$. Thus, $$\mid E_d\mid =$$ infinity at the y-intercept.

At the midpoint, the price elasticity of demand is equal to negative one.

$E_d = \frac{\%ΔQ^d}{\%ΔP} = \left(\frac{ΔQ^d}{ΔP}\right)\left(\frac{P}{Q^d}\right) = -1 \label{1.17}$

At the midpoint, the slope of the demand curve is equal to minus one $$(\dfrac{ΔQ^d}{ΔP} = 1)$$, and the price is equal to the quantity demanded $$(P = Q^d)$$. Therefore the own price elasticity of demand at the midpoint of a linear demand curve is equal to minus one $$\left(\dfrac{ΔQ^d}{ΔP}\right)\left(\dfrac{P}{Q_d}\right) = -1$$.

A valuable lesson is learned in this example: be careful to distinguish between the slope of a demand curve and the elasticity of demand. When interpreting graphs, the slope is not a good determinant of elasticity, since a graph could be drawn steep or shallow depending on the units. The elasticity is related to the slope, but it is not equal to the slope!

## Agricultural Policy Example of Elasticity of Demand

The impact of agricultural policies depends critically on the elasticity of demand. It was claimed earlier that the price elasticity of demand is the most important thing that a business firm can know. This section provides evidence of the importance of the price elasticity of demand. The own price elasticity of demand for food is inelastic in a domestic economy with no trade. Everyone must eat, and the caloric intake will not be greatly influenced by the price of food.

This changes enormously in a global economy. In an open economy that has international trade, there are many overseas customers for food exports, and many competing nations that export food. For example, the US is a major wheat exporter. Other wheat exporting nations include: Canada, Australia, Argentina, the European Union (EU), and many of the former Soviet nations in Eastern Europe such as Ukraine. In this case, the US faces a highly elastic demand for wheat in the global economy: if the US increased the price of wheat above the world price, wheat importers would shift purchases from the US to other wheat exporters. A global economy changes the effectiveness of price policies enormously.

Prior to 1972, the United States agricultural sector could be characterized as a domestic economy, with less food and agricultural trade. In this case, the demand for food was primarily domestic, and thus relatively inelastic (Figure $$\PageIndex{8}$$, left panel).

Starting in 1933, agricultural price supports increased the price of wheat above the market equilibrium level. This policy worked well, as long as the surplus was eliminated. One way to eliminate the surplus was through acreage restrictions, which limited the number of acres planted to wheat ($$ΔQ$$ in the left panel of Figure $$\PageIndex{8}$$). Acreage limitations and production quotas were used to decrease the quantity of wheat in the market. These policies worked well prior to 1972, since the US agricultural economy was primarily domestic, characterized by an inelastic demand curve. The decrease in quantity led to a larger increase in price $$(ΔP > ΔQ^d)$$, given the inelastic demand in a domestic economy.

In 1972, major changes in international exchange rate policies, together with poor weather in Asia, led to the globalization of US food and agricultural markets. A larger percentage of the US wheat crop was exported, and the inelastic demand that prevailed prior to 1972 became more elastic in the globalized environment (right panel, Figure $$\PageIndex{8}$$). Although the wheat market became globalized, the policies did not. During the 1980s, the US maintained price supports and production controls in the seven basic commodities (defined by the USDA as: wheat, corn, sorghum, sugar, cotton, rice, and tobacco). These policies were counterproductive, as they priced the grain out of the world market. The US attempted to increase the market share of wheat trade, only to find that the US price was higher than the other major wheat exporters.

Figure $$\PageIndex{8}$$ shows why the policies implemented in 1933 were hurting more than they were helping. Production controls were decreasing the quantity of wheat. In the domestic economy (left panel of Figure $$\PageIndex{8}$$, pre-1970), this achieved the objectives of the policies: wheat producer were made better off, since the increase in price was greater than the decrease in quantity. This all changed in the globalized world after 1972 (right panel of Figure $$\PageIndex{8}$$, post 1972). With an elastic demand, the decrease in quantity did not result in large price increases. Price supports raised the US price above the world price of wheat. These policies were not working, and in 1996, they were changed to make the US grain industry more competitive in the global market. Today, a large fraction of all grain produced in the US is exported.

In summary, policies intended to help producers have greatly divergent outcomes, depending on the price elasticity of demand. In a domestic economy, the demand for food and agricultural products is typically inelastic. In this case, production controls and price supports will achieve the policy goal of helping producers: the price increases will be larger than the quantity decreases (left panel, Figure $$\PageIndex{8}$$). In a global economy, the demand for food and agricultural goods is elastic: there are many nations that export grains (right panel, Figure $$\PageIndex{8}$$). In this case the policy that helps producers the most is technological change, which will shift the supply curve to the right. With an elastic demand, the increase in quantity is larger than the decrease in price.

This is the same strategy that Walmart utilizes: everyday low prices. Sam Walton found that the increase in sales due to low prices more than offset the decrease in price $$(ΔQ^d > ΔP)$$. This is true in any market characterized by an elastic demand. Since most consumer goods in the United States have many substitutes, Walton’s lower-price strategy led to Walmart becoming the most successful retailer in the history of the world.

## Calculation of Market Supply and Demand Elasticities

In Section 1.3.4 above, inverse supply and demand curves were used to calculate the equilibrium price and quantity of phones.

The inverse demand and supply functions were:

$P = 100 – 2Q^d \label{1.18}$

$P = 20 + 2Q^s \label{1.19}$

Where $$P$$ is the price of phones in USD/unit, and $$Q$$ is the quantity of phones in millions. By setting the two equations equal to each other, the intersection of the inverse supply and demand curves was found, yielding the equilibrium market price and quantity:

$P^e = \text{ USD } 60\text{/phones,}$

and

$Q^e = 20 \text{ million phones.}$

The graph of the phone market is replicated in Figure $$\PageIndex{9}$$.

To calculate the own price elasticities of supply and demand, simple calculus provides an easy solution. Recall the definition of the own price elasticity of demand:

$E_d = \frac{\%ΔQ^d}{\%ΔP} = \left(\frac{ΔQ_d}{ΔP}\right)\left(\frac{P}{Q_d}\right) = \left(\frac{∂Q_d}{∂P}\right)\left(\frac{P}{Q_d}\right)\label{1.20}$

The delta sign $$(Δ)$$ refers to a small change in a variable. This is the same as the derivative sign, “$$∂$$.” The difference is that the derivative indicates in infinitesimally small change, whereas the delta sign is a discrete change, which is the same idea, just larger. Therefore, the delta signs can be replaced with the derivative signs in the equation that defines the price elasticity of demand. For example, the slope of a function is $$\dfrac{Δy}{Δx}$$. The slope of the function at a given point on the function is $$\dfrac{∂y}{∂x}$$.

The last expression in the equation shows that to calculate $$E_d$$, use the derivative of quantity with respect to price, and the levels of $$P$$ and $$Q$$. At the equilibrium point, the equilibrium levels of $$P$$ and $$Q$$ are known. To find the derivative, begin by taking the derivative of the inverse demand equation (Equation \ref{1.18}).

$\frac{∂P}{∂Q^d} = – 2\label{1.21}$

This is simply the power function rule from calculus [if $$y =ax^b$$, then $$\dfrac{∂y}{∂x} = abx^{b-1}$$]. Notice something important: the derivative of the inverse demand equation is the inverse of what is needed to calculate the price elasticity. This is due to the inverse demand function being, “price-dependent,” with $$P$$ on the left hand side. To find the derivative $$\dfrac{∂Q^d}{∂P}$$, invert the derivative by dividing one by the derivative.

$\frac{∂Q^d}{∂P} = –\left(\frac{1}{2}\right) \label{1.22}$

$E_d = \frac{\%ΔQ^d}{\%ΔP} = \left(\frac{ΔQ^d}{ΔP}\right)\left(\frac{P}{Q^d}\right) = \left(\frac{∂Q^d}{∂P}\right)\left(\frac{P}{Q^d}\right) = -\left(\frac{1}{2}\right)\left(\frac{60}{20}\right) = -1.5\label{1.23}$

The absolute value of the own price elasticity of demand at the equilibrium point is:

$$\mid E_d\mid = 1.5$$. The demand for phones is elastic: if the price were increased one percent, the decrease in phone purchases would be 1.5 percent.

The own price elasticity of supply can be found using the same procedure:

$E_s = \frac{\%ΔQ^s}{\%ΔP} = \left(\frac{ΔQ^s}{ΔP}\right)\left(\frac{P}{Q^s}\right) = \left(\frac{∂Q^s}{∂P}\right)\left(\frac{P}{Q^s}\right)\label{1.24}$

First, take the derivative of the inverse supply function (Equation \ref{1.19}).

$\frac{∂P}{∂Q^s} = + 2 \label{1.25}$

Invert this derivative to find the derivative needed to calculate the price elasticity of supply:

$\frac{∂Q^s}{∂P} = + \left(\frac{1}{2}\right). \label{1.26}$

Then plug in the ingredients of the own price elasticity of supply:

$E_s = \frac{\%ΔQ^s}{\%ΔP} = \left(\frac{ΔQ^s}{ΔP}\right)\left(\frac{P}{Q^s}\right) = \left(\frac{∂Q^s}{∂P}\right)\left(\frac{P}{Q^s}\right) = \left(\frac{1}{2}\right)\left(\frac{60}{20}\right) = 1.5 \label{1.27}$

The price elasticity of supply is also elastic: a one percent increase in price results in a 1.5 percent increase in the quantity supplied of phones.

Two points are worth mentioning here. First, the price elasticities of supply and demand are not always symmetrical, as they are in this case (-1.5 and +1.5). The elasticities depend on the shape of the inverse supply and demand functions. The symmetry of the functions used here can be seen in Figure $$\PageIndex{9}$$. When the inverse supply and demand functions are not symmetrical, the absolute values of the elasticities will be of different magnitudes. The second important point concerns the use of inverse supply and demand functions. The inverse functions are used to align with the “backwards” nature of the supply and demand graphs: price is the independent variable, but appears on the vertical axis. To find the derivative needed to calculate the price elasticities, the procedure above first took the derivative of the inverse function, then inverted it to achieve $$\dfrac{∂Q}{∂P}$$. This derivative could also be found be first, inverting the inverse function to get the quantity isolated on the left hand side, then taking the derivative. This alternative procedure will result in the same elasticity calculation as the one used above. To test your knowledge, try this procedure to double check your answers!

This page titled 1.4: Welfare Economics - Consumer and Producer Surplus is shared under a CC BY-NC license and was authored, remixed, and/or curated by Andrew Barkley (New Prairie Press/Kansas State University Libraries) .

format: plain


text:
# Q12

Number Theory Level pending

Let x be the elements of the set A={1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120} and $${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 }$$ be positive integers and d be the number of integral solutions of $${ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }=x$$ then d is divisible by

×


text:
# Can you relate it to area of triangle?

Geometry Level 4

If $$P$$ be any interior point of the equilateral triangle $$ABC$$ of side length 2 units and also $${ x }_{ a },{ x }_{ b },{ x }_{ c }$$ be the distances of P from the sides $$BC,CA$$ and $$AB$$ respectively. Compute $${ x }_{ a }+{ x }_{ b } + {x }_{ c }$$.

×


text:
User: Paul Brennan

Paul Brennan10
Reputation:
10
Status:
New User
Location:
Cardiff, UK
Website:
http://rforbiochemists...
brennanpcardiff
Last seen:
8 months, 2 weeks ago
Joined:
1 year, 11 months ago
Email:
b****************@gmail.com

Reader in School of Medicine, Cardiff University, Cardiff, UK.

Interest: data visualisation, R, biochemistry, open data, equality & diversity, education.

Creator of R for Biochemists Blog

Posts by Paul Brennan

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... Hi James, Nice job. To add to your answer, we can use the amino acid sequence to calculate the protein length. Here is some code that will do that. Best wishes, Paul ` # Load the package required to read JSON files. library("rjson") url<-"https://www.ebi.ac.uk/proteins/api/featu ...
written 8 months ago by Paul Brennan10
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... Hello Tiphaine, Sorry for the slow response to this question. Yes you can easily add new features with your choice of colors and they can be added below or above the protein. There is some code here that shows how to plot variants: https://rforbiochemists.blogspot.com/2018/02/my- ...
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... Another check of my man files revealed too long a title generated by roxygen2 documentation.  It was in a data object and the title went over two lines.  ...
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... Hi,       I'm BiocChecking my package and through various ways, I'm down to a single NOTE that is created by a single line of code. Here is the note.  * Checking formatting of DESCRIPTION, NAMESPACE, man pages, R source, and vignette source... * NOTE: Consider shorter lines; 1 lines (0%) ar ...
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... Thanks for the prompt, helpful and correct response. One less warning.  In my case it is a repeat of the @format tag... You left out the @format tag is that OK? Thanks again,  Paul ...
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... Hello,       I am developing a package to automate the drawing of protein schematics. I don't think a similar package exists in R or Bioconductor and I hope I haven't missed someone else's package. The github repo is here: https://github.com/brennanpincardiff/drawProteins. I'm working to raise it t ...
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# is the inequality true??

It is given that $f$ is a continuous function and $A$ a bounded set.

Is the inequality $$\sup_{x \in A} \int_a^b f(x) dx \leq \int_a^b sup_{x \in A} f(x) dx$$ true?

If it is true,could we maybe prove it,with this relation: $$\sup_{x \in A} (\sup_{x \in A} f(x))=\sup_{x \in A} f(x)$$ ?

• the expression on the left side is ill-formed. x can not be a variable of the sup and a variable of the integral at the same time. – Michael Apr 30 '14 at 19:08
• Yes,right..but $\sup_{x \in A} f(x)$ is a real number,so it does not depend from $x$,or not? – evinda Apr 30 '14 at 19:10
• @Michael It's not ill-formed, just weird. The integral operation re-binds $x$, i.e. introduces a new variable that happens to also be called $x$, and within the integral $x$ refers to that new variable. The outter variable $x$ - the one bound by the supremem - is therefore simply unused. – fgp Apr 30 '14 at 19:15
• I think he meant $\sup_{a,b \in A}$ on the LHS – PA6OTA Apr 30 '14 at 19:20

The statement is false.

Counterexample:

$$a=0, b=1, f(x)=x, A=(-3,-2)$$

Because: Who said that $A$ has to be the integration intervall?

As states, this is false. You need to impose some restriction on $a,b$ vs. $A$ for this to hold, otherwise you can do something like $a=0$, $b=2$, $A=[0,1]$ and let $f(x) = 0$ if $x \leq 1$ and $1$ otherwise. Then $$\int_a^b f = 1 > \int_a^b \sup_{x\in[0,1]} f(x) = \int_a^b 0 = 0 \text{.}$$

Also, note that $x$ isn't a free variable in the expression $\int_a^b f(x)\,dx$, so writing $$\sup_{x \in A} \int_a^b f(x)\,dx$$ is a bit weird. It comes down to writing $\sup_{x \in A} c$ for some constant $c$, which is of course just $c$.

However, you do have that $$\int_A f(x) \,dx \leq \int_A |f(x)| \,dx \leq \int_A \sup_{x\in A}|f(x)| \,dx = |A|\cdot \sup_{x\in A}|f(x)| \text{,}$$ where $|A|$ denoted the measure of $A$, i.e. $|A| = \int_A 1 \,dx$.

You also have that if $f(x) < g(x)$ then $$\int_A f(x) \,dx \leq \int_A g(x) \,dx \text{.}$$ For $f \geq 0$ this is clear. In the general case (this is where I was confused prevously), you can split $f$ into it's positive and negative parts, i.e. write $f(x) = f^+(x) - f^-(x)$ where $f^+,f^- \geq 0$, and the same for $g$. Then, $f^+ \leq g^+$ and $f^- \geq g^-$, and thus $$\int_A g(x) - \int_A f(x) = \underbrace{\int_A g^+ - \int_A f^+}_{\geq 0} + \underbrace{\int_A f^- - \int_A g^-}_{\geq 0} > 0$$

• But f is continous... – rlartiga Apr 30 '14 at 19:14
• Isn't it $$\sup_{x \in [0,1]} f(x) dx=0$$? – evinda Apr 30 '14 at 19:16
• really... $\int_0^1 \sup_{x\in[0,1]} f(x) \,dx = 0$ !!! – Michael Apr 30 '14 at 19:20
• @Michael Yeah, ups, sorry for that. – fgp Apr 30 '14 at 19:21


text:
### On efficient learning algorithms for neural networks.

FieldValue
dc.contributor.authorGolea, Mostefa.
dc.date.accessioned2009-03-23T14:12:26Z
dc.date.available2009-03-23T14:12:26Z
dc.date.created1993
dc.date.issued1993
dc.identifier.citationSource: Dissertation Abstracts International, Volume: 54-09, Section: B, page: 4731.
dc.identifier.isbn9780315825185
dc.identifier.urihttp://hdl.handle.net/10393/6508
dc.identifier.urihttp://dx.doi.org/10.20381/ruor-11313
dc.description.abstractInductive Inference Learning can be described in terms of finding a good approximation to some unknown classification rule f, based on a pre-classified set of training examples $\langle$x,f(x)$\rangle.$ One particular class of learning systems that has attracted much attention recently is the class of neural networks. But despite the excitement generated by neural networks, learning in these systems has proven to be a difficult task. In this thesis, we investigate different ways and means to overcome the difficulty of training feedforward neural networks. Our goal is to come up with efficient learning algorithms for new classes (or architectures) of neural nets. In the first approach, we relax the constraint of fixed architecture adopted by most neural learning algorithms. We describe two constructive learning algorithms for two-layer and tree-like networks. In the second approach, we adopt the "probably approximately correct" (PAC) learning model and we look for positive learnability results by restricting the distribution generating the training examples, the connectivity of the networks, and/or the weight values. This enables us to identify new classes of neural networks that are efficiently learnable in the chosen setting. In the third and final approach, we look at the problem of learning in neural networks from the average case point of view. In particular, we investigate the average case behavior of the well known clipped Hebb rule when learning different neural networks with binary weights. The arguments given for the "efficient learnability" range from extensive simulations to rigorous mathematical proofs.
dc.format.extent196 p.


text: Draw a sketch of the graph of $f(x)$ for $a\leq x \leq b$ and interpret from your diagram $bf(b) - af(a).$


text:
# Bash script to calculate when I wake up

This script tries to determine the time when you woke up.

Of course, it only applies if you turn your computer on not too long ago after getting out of bed.

If not, it might still be useful as an indicator of when you turned your computer on.

It works by getting information from journalctrl, sleep-resume dates and boot dates.

It then loops through them and tries to find when the difference is above a certain gap (5 hours by default, it can be configured with the -g flag).

Then it prints the time and "timeago" information of when you probably woke up.

#!/bin/bash

gap=18000

while [ ! $# -eq 0 ] do case "$1" in
--help | -h)
printf "Version: 1.2.1\nFlags:\n\t-g or --gap: Specifies the chunks of time between two dates to determine when you probably woke up (default is 5)\n"
exit
;;
--gap | -g)
gap=$(($2 * 3600))
;;
esac
shift
done

current_date=$(date +%s) readarray -t sleep_dates < <(journalctl -o short-unix -t systemd-sleep | grep resumed | tail -50 | awk -F. '{print$1}')
readarray -t boot_dates < <(journalctl --list-boots | tail -50 | awk '{ d2ts="date -d \""$3" "$4" " $5"\" +%s"; d2ts | getline$(NF+1); close(d2ts)} 1' | awk 'NF>1{print $NF}') dates=( "${sleep_dates[@]}" "${boot_dates[@]}" ) readarray -t sorted_dates < <(printf '%s\n' "${dates[@]}" | sort)

for (( i=${#sorted_dates[@]}-1 ; i>=0; i-- )); do diff=$((sorted_dates[i] - sorted_dates[i - 1]))
diff2=$((sorted_dates[i - 1] - sorted_dates[i - 2])) if [ "$diff" -gt "$gap" ] && [ "$diff2" -lt "$gap" ]; then sdate=$(date --date @${sorted_dates[i]} +"%r") diff2=$((current_date - sorted_dates[i]))
hours_ago=$(echo "scale=2;${diff2}/3600" | bc)
whole_hours=$(echo "(${hours_ago})/1" | bc)
decimals=$(echo "${hours_ago}" | grep -Eo "\.[0-9]+$") minutes_ago=$(echo "(${decimals}*60)/1" | bc) if [ "$whole_hours" -eq "1" ];
then
shours="hour"
else
shours="hours"
fi

if [ "$minutes_ago" -eq "1" ]; then sminutes="minute" else sminutes="minutes" fi message="${whole_hours} ${shours} and${minutes_ago} ${sminutes} ago (${sdate} )"
echo "$message" break fi done  ## 2 Answers The thing with bash is more often then not... there is a tool that you can leverage to do what you want. When writing bash scripts, I try write as little bash as possible, and to leverage these tools where best possible to avoid bash (seems counter intuitive, but is the nature of bash to write as little bash as possible). For example have you heard of the command tuptime? (https://manpages.debian.org/testing/tuptime/tuptime.1.en.html) This is what it looks like after a reboot (is packaged with apt): $ sudo apt-get install tuptime
$tuptime | grep "System downtime" System downtime: 0.15 % - 32 seconds  Is not a complete answer because does not cover the >5 hrs case above. For reference, here is a run of tuptime $ tuptime
System startups:    2   since   11:02:34 AM 02/05/2020
System shutdowns:   1 ok   <-   0 bad
System uptime:      99.85 %   -   5 hours, 56 minutes and 45 seconds
System downtime:    0.15 %   -   32 seconds
System life:        5 hours, 57 minutes and 17 seconds

Largest uptime:     5 hours, 45 minutes and 12 seconds   from   11:02:34 AM 02/05/2020
Shortest uptime:    11 minutes and 33 seconds   from   04:48:18 PM 02/05/2020
Average uptime:     2 hours, 58 minutes and 23 seconds

Largest downtime:   32 seconds   from   04:47:46 PM 02/05/2020
Shortest downtime:  32 seconds   from   04:47:46 PM 02/05/2020
Average downtime:   32 seconds

Current uptime:     11 minutes and 33 seconds   since   04:48:18 PM 02/05/2020

If wanting something more specific, I would also consider last command which gives you information about when people logged in or out (using http://man7.org/linux/man-pages/man1/last.1.html).

$last reboot --time-format iso reboot system boot 4.15.0-76-generi 2020-02-05T16:48:22-0800 still running reboot system boot 4.15.0-76-generi 2020-02-05T11:02:38-0800 - 2020-02-05T16:47:57-0800 (05:45) reboot system boot 4.15.0-76-generi 2020-02-04T14:30:53-0800 - 2020-02-05T16:47:57-0800 (1+02:17) reboot system boot 4.15.0-76-generi 2020-02-03T15:28:38-0800 - 2020-02-04T14:30:29-0800 (23:01) reboot system boot 4.15.0-76-generi 2020-02-03T10:55:02-0800 - 2020-02-03T15:28:13-0800 (04:33)  # Philosophy You seemed to worry about the usefulness of this a bit when you said "it might still be useful..." It doesn't have to be useful! Was it fun? Did you learn something? I love personal data collection projects like this. It is amazing what it can lead to. I also find it interesting to see the idea of "write as little bash as possible" in a code review. While shell scripting isn't the oasis in a technology desert that it was thirty years ago it is still a good way to get a variety of things done. I write python for my job but I often have bash wrapper scripts. Each language has its strengths and the more you learn about each of them the better general programmer you will be. And if you ignore what you should be doing in bash then it is amazing what you can do. # Good Parts Yes, really, I'm going to review your code.... Starting with some things I like: • good indentation • decent variable names • using $( for command substitution
• quoting defensively

# Suggestions

• Using [[ for conditionals is a best practice and will help you avoid some surprises. "With double square brackets you don’t need to escape parenthesis and unquoted variables work just fine even if they contain spaces (meaning no word splitting or glob expansion)." source
• You don't need the inner square brackets in if [ "$diff" -gt "$gap" ] && [ "$diff2" -lt "$gap" ]; then . Combining that with the previous point would lead to if [[ "$diff" -gt "$gap" && "$diff2" -lt "$gap" ]]; then
• Use a heredoc for your usage print-out so it isn't one long line. It will be much easier to maintain and expand your help info in a heredoc.
• For the code

if [ "\$whole_hours" -eq "1" ];
then

you can put the then up next to the ; or just drop the ; since the line ending will end the if command. Also, quoting the 1 is unnecessary, but harmless.

• Using #!/usr/bin/env bash in your shebang line might be a good idea.


text:
PolarCorrection¶

class sharpy.generators.polaraeroforces.PolarCorrection[source]

This generator corrects the aerodynamic forces from UVLM based on the airfoil polars provided by the user in the aero.h5 file. Polars are entered for each airfoil, in a table comprising AoA (rad), CL, CD, CM.

This generator_id = 'PolarCorrection' and can be used in the coupled solvers through the correct_forces_method setting as:

These are the steps needed to correct the forces:

1. The force coming from UVLM is divided into induced drag (parallel to the incoming flow velocity) and lift

(the remaining force).

If cd_from_cl == 'on'.
1. The viscous drag and pitching moment are found at the computed lift coefficient. Then forces and moments are updated

Else, the angle of attack is computed:

1. The angle of attack is computed based on that lift force and the angle of zero lift computed from the airfoil polar and assuming the potential flow lift curve slope of $$2 \pi$$

1. The drag force is computed based on the angle of attack and the polars provided by the user

2. If correct_lift == 'on', the lift coefficient is also corrected with the polar data. Else, only the UVLM results are used.

The pitching moment is added in a similar manner as the viscous drag. However, if moment_from_polar == 'on' and correct_lift == 'on', the total moment (the one used for the FSI) is computed just from polar data, overriding any moment computed in SHARPy. That is, the moment will include the polar pitching moment, and moments due to lift and drag computed from the polar data.

This generator takes in the following settings.

Name

Type

Description

Default

correct_lift

bool

Correct lift according to the polars

False

cd_from_cl

bool

Interpolate the C_D for the given C_L, as opposed to getting the C_D from the section AoA.

False

moment_from_polar

bool

If correct_lift is selected, it will compute the pitching moment simply from polar derived data, i.e. the polars Cm and the momentsarising from the lift and drag (derived from the polar) contribution. Else, it will add the polar Cm to the moment already computed by SHARPy.

False

generate(**params)[source]
Keyword Arguments
Returns

New corrected structural forces

Return type

np.array


text:
# Help Single slit diffraction

1. Apr 27, 2005

### six789

Help!! Single slit diffraction

heya all... this is the problem..
Predict whether violet light(lambda=404nm) or red light (lambda=702nm) will have a wider central maximum when used to generate a single-slit diffraction pattern. Calculate the difference if the light is incident on a 6.9x10^-5 wide slit falling onot a screen 85cm away..

i know that the red light will have a wider central maximum, but i dont know the unknown, is it Ym from the formula Ym=(m*lambda*L)/W? and i dont know the value for m, is it 5 since the gap between the red light and the violet light is 5... im not sure... and the value of red light in my book is 7.3x10^-3m... HELP me please...im soo confused..

THANKS SO MUCH in advance for the big help.

Last edited: Apr 27, 2005
2. Apr 27, 2005

### six789

3. Apr 27, 2005

### xanthym

For single-slit diffraction, the first MINIMUM will occur at (m=1) of your formula:

$$1: \ \ \ \ Y_{1} \ \, = \ \, \frac{\color{red}\mathbf{(1)}\color{black}\lambda L}{W}$$

where "L" is the distance to screen, and "W" the slit width. The "length" of the Central Maximum measured from the first minimum below the Central Peak to the first minimum above the Central Peak is (2*Y1). The above formula applies to each different "λ" you wish to evaluate.

~~

Last edited: Apr 27, 2005
4. Apr 27, 2005

### six789

im so confused... wat is the unknown here?? is it L or Y?

5. Apr 27, 2005

### xanthym

The unknown is "Y". You know the value of "L", which is the distance from the slit to the screen, given in the problem to be (L = 0.85 meters).

~~

Last edited: Apr 27, 2005
6. Apr 27, 2005

### six789

and the value for W is 6.9x10^-5?

7. Apr 27, 2005

### xanthym

You did not include units for the above value. Assuming the units to be "meters", then yes you're correct that {W = 6.9x10^(-5) meters}.

~~

Last edited: Apr 27, 2005
8. Apr 27, 2005

### six789

yes, thats what i did.. but why is my answer incorrect to the answer in the book... it should be 7.3x10^-3m for the red light..

9. Apr 27, 2005

### xanthym

According to your msg, problem requires calculation of the DIFFERENCE in Central Maxima "lengths" for Red & Violet light:

$$2: \ \ \ \ \mbox{Problem Requirement} \ \, = \ \, \color{red}2Y_{1}(Red) \ \, - \ \, 2Y_{1}(Violet)$$

Recalculate your results using above Eq #2.

~~

Last edited: Apr 27, 2005
10. Apr 27, 2005

### six789

i just have to multiply my answers to 2 or ...??

11. Apr 27, 2005

### six789

yeah i got teh right answer...
thanks so much man..

12. Apr 27, 2005

### six789

but why do u have to do that .. i mean multiply it to 2??

13. Apr 27, 2005

### xanthym

Use Eq #2 given in Msg #9. The value for each Y1 must be multiplied by 2 to obtain the FULL "length" of the Central Maximum. The Central Maximum "length" must be calculated for BOTH red and violet light, and then placed into Eq #2 given in Msg #9.

~~

Last edited: Apr 27, 2005
14. Apr 27, 2005

### six789

oh now i see.. thanks again man..

15. Apr 27, 2005

### six789

Commercial satellites are able to resolve objects separated by only 1.0m. If those satellites orbit the earth at an altitude of 650km, determine the sixe of the satellites' circular imaging aperture. Use 455nm light for the light in the lenses of the satellites..
can u help me again... i dont know waht formula to use here??

16. Apr 27, 2005

### xanthym

The formula for circular aperture diffraction is different from that for single-slit diffraction. The minimum resolvable distance "Y" at distance "L" from a circular aperture of diameter "W" using light of wavelength "λ" is given by:

$$3: \ \ \ \ Y \ \, = \ \, \frac{\color{red}\mathbf{(1.22)}\color{black}\lambda L}{W}$$

Solve for aperture diameter "W" (in meters) using other variable values given by problem {Y=(1 meter), L=(650e3 meters), λ=(455e(-9) meters)}.

~~

Last edited: Apr 27, 2005
17. Apr 28, 2005

### six789

where did u get the formula for this?? i just know the formula... tetha = 1.22(lambda)/D... can u tell me how u derived from that formula??

Last edited: Apr 28, 2005
18. Apr 29, 2005

### xanthym

$$\setlength{\unitlength}{.001cm} \begin{picture}(6000,5000)(500,-7000) \color{red} \thicklines \linethickness{1pt} \put(6001,-2161){\line(-1,-1){600}} \put(5401,-2761){\line( 1,-1){600}} \put(6001,-661){\line( 0,-1){4500}} \put(1201,-2761){\line( 1, 0){4800}} \put(4201,-661){\line( 0,-1){1800}} \put(6001,-3361){\line(-3, 1){4815}} \put(4201,-3061){\line( 0,-1){2100}} \put(1201,-3350){A} \put(1201,-1561){B} \put(6151,-2950){C} \put(6151,-2300){\color{blue} \large Triangle Represents Point A's Airy Diffraction Central Max} \put(7500,-3300){\color{blue}\theta \, = \, \angle CED \, = \, \angle AEB} \put(7500,-4400){\color{blue} \frac{Length(AB)}{Length(AE)} \, \approx \, \theta \, \approx \, \frac{(1.22)\lambda}{(Aperture \ Diameter)} } \put(6151,-3600){\mbox{D}} \put(4351,-2611){\mbox{E}} \put(0,-7000){\mbox{.}} \end{picture}$$
You are certainly correct that several approximations are involved. At large distances, the approximations shown above are usually valid. The Triangle represents Point "A"s Airy Diffraction resulting from the Circular Aperture at Point "E". Point "B" can be resolved from Point "A" by the Rayleigh Criteria when Point "B"s Airy Diffraction Central MAX falls on Point "A"s First MIN. This occurs at angle "θ" shown above. The other relationships can be determined from the formulas shown.

Derivation and discussion of Airy Diffraction from a circular aperture (a special case of Fraunhofer Diffraction) can be found here:
http://scienceworld.wolfram.com/physics/FraunhoferDiffractionCircularAperture.html
A brief overview discussion can be found here:
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp.html

~~

Last edited: Apr 29, 2005
19. Apr 29, 2005

### six789

xanthym, i know what u mean... but im not familiar with the formula u said...
so u mean.. 1.22 is the m?

Last edited: Apr 29, 2005
20. Apr 29, 2005

### xanthym

Last edited: Apr 29, 2005


text:
## College Physics (4th Edition)

The distance between the source and the detector is $80~km$
We can use the speed $v_1$ and the travel time $t_1$ of the longitudinal waves to write an expression for the distance: $d = v_1~t_1$ Note that $t_2 = t_1+2.0~s$. We can use the speed $v_2$ and the travel time $t_2$ of the transverse waves to write an expression for the distance: $d = v_2~t_2 = v_2~(t_1+2.0~s)$ Since the distance is the same, we can equate the two equations to find $t_1$: $v_1~t_1 = v_2~(t_1+2.0~s)$ $(v_1-v_2)~t_1 = (2.0~s)~v_2$ $t_1 = \frac{(2.0~s)~v_2}{v_1-v_2}$ $t_1 = \frac{(2.0~s)(8.0~km/s)}{10.0~km/s-8.0~km/s}$ $t_1 = 8.0~s$ We can find the distance $d$: $d = v_1~t_1 = (10.0~km/s)(8.0~s) = 80~km$ The distance between the source and the detector is $80~km$.


text:
• Chowdhury F-U-Z

Articles written in Pramana – Journal of Physics

• A model of evaluating the pseudogap temperature for high-temperature superconductors

We have presented a model of evaluating the pseudogap temperature for high temperature superconductors using paraconductivity approach. The theoretical analysis is based on the crossing point technique of the conductivity expressions. The pseudogap temperature T $^∗$ is found to depend on dimension and is calculated for 2D and 3D superconducting samples. Numerical calculation is given in favour of the YBCO and doped SmFeAsO$_{1−x}$ samples.

• # Pramana – Journal of Physics

Volume 96, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Reference documentation for deal.II version Git 99efdf013c 2021-02-27 20:52:41 -0500
Triangulation< dim, spacedim >::Signals Struct Reference

#include <deal.II/grid/tria.h>

## Public Attributes

boost::signals2::signal< void()> create

boost::signals2::signal< void()> pre_refinement

boost::signals2::signal< void()> post_refinement

boost::signals2::signal< void()> pre_partition

boost::signals2::signal< void()> mesh_movement

boost::signals2::signal< void(const typename Triangulation< dim, spacedim >::cell_iterator &cell)> pre_coarsening_on_cell

boost::signals2::signal< void(const typename Triangulation< dim, spacedim >::cell_iterator &cell)> post_refinement_on_cell

boost::signals2::signal< void(const Triangulation< dim, spacedim > &destination_tria)> copy

boost::signals2::signal< void()> clear

boost::signals2::signal< void()> any_change

boost::signals2::signal< unsigned int(const cell_iterator &, const CellStatus), CellWeightSum< unsigned int > > cell_weight

boost::signals2::signal< void()> pre_distributed_refinement

boost::signals2::signal< void()> post_distributed_refinement

boost::signals2::signal< void()> pre_distributed_repartition

boost::signals2::signal< void()> post_distributed_repartition

boost::signals2::signal< void()> pre_distributed_save

boost::signals2::signal< void()> post_distributed_save

## Detailed Description

### template<int dim, int spacedim = dim> struct Triangulation< dim, spacedim >::Signals

A structure that has boost::signal objects for a number of actions that a triangulation can do to itself. Please refer to the "Getting notice when a triangulation changes" section in the general documentation of the Triangulation class for more information and examples.

For documentation on signals, see http://www.boost.org/doc/libs/release/libs/signals2 .

Definition at line 2071 of file tria.h.

## ◆ create

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::create

This signal is triggered whenever the Triangulation::create_triangulation or Triangulation::copy_triangulation() is called. This signal is also triggered when loading a triangulation from an archive via Triangulation::load().

Definition at line 2080 of file tria.h.

## ◆ pre_refinement

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_refinement

This signal is triggered at the beginning of execution of the Triangulation::execute_coarsening_and_refinement() function (which is itself called by other functions such as Triangulation::refine_global() ). At the time this signal is triggered, the triangulation is still unchanged.

Definition at line 2089 of file tria.h.

## ◆ post_refinement

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::post_refinement

This signal is triggered at the end of execution of the Triangulation::execute_coarsening_and_refinement() function when the triangulation has reached its final state.

Definition at line 2096 of file tria.h.

## ◆ pre_partition

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_partition

This signal is triggered at the beginning of execution of the GridTools::partition_triangulation() and GridTools::partition_triangulation_zorder() functions. At the time this signal is triggered, the triangulation is still unchanged.

Definition at line 2104 of file tria.h.

## ◆ mesh_movement

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::mesh_movement

This signal is triggered when a function in deal.II moves the grid points of a mesh, e.g. GridTools::transform. Unfortunately, modification of a vertex in user code through cell_iterator->vertex(v) = xxxx cannot be detected by this method.

Definition at line 2113 of file tria.h.

## ◆ pre_coarsening_on_cell

template<int dim, int spacedim = dim>
 boost::signals2::signal::cell_iterator &cell)> Triangulation< dim, spacedim >::Signals::pre_coarsening_on_cell

This signal is triggered for each cell that is going to be coarsened.

Note
This signal is triggered with the immediate parent cell of a set of active cells as argument. The children of this parent cell will subsequently be coarsened away.

Definition at line 2124 of file tria.h.

## ◆ post_refinement_on_cell

template<int dim, int spacedim = dim>
 boost::signals2::signal::cell_iterator &cell)> Triangulation< dim, spacedim >::Signals::post_refinement_on_cell

This signal is triggered for each cell that just has been refined.

Note
The signal parameter cell corresponds to the immediate parent cell of a set of newly created active cells.

Definition at line 2134 of file tria.h.

## ◆ copy

template<int dim, int spacedim = dim>
 boost::signals2::signal &destination_tria)> Triangulation< dim, spacedim >::Signals::copy

This signal is triggered whenever the triangulation owning the signal is copied by another triangulation using Triangulation::copy_triangulation() (i.e. it is triggered on the old triangulation, but the new one is passed as an argument).

Definition at line 2144 of file tria.h.

## ◆ clear

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::clear

This signal is triggered whenever the Triangulation::clear() function is called and in the destructor of the triangulation. This signal is also triggered when loading a triangulation from an archive via Triangulation::load() as the previous content of the triangulation is first destroyed.

The signal is triggered before the data structures of the triangulation are destroyed. In other words, the functions attached to this signal get a last look at the triangulation, for example to save information stored as part of the triangulation.

Definition at line 2159 of file tria.h.

## ◆ any_change

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::any_change

This is a catch-all signal that is triggered whenever the create, post_refinement, or clear signals are triggered. In effect, it can be used to indicate to an object connected to the signal that the triangulation has been changed, whatever the exact cause of the change.

Note
The cell-level signals pre_coarsening_on_cell and post_refinement_on_cell are not connected to this signal.

Definition at line 2170 of file tria.h.

## ◆ cell_weight

template<int dim, int spacedim = dim>
 boost::signals2::signal > Triangulation< dim, spacedim >::Signals::cell_weight

This signal is triggered for each cell during every automatic or manual repartitioning. This signal is somewhat special in that it is only triggered for distributed parallel calculations and only if functions are connected to it. It is intended to allow a weighted repartitioning of the domain to balance the computational load across processes in a different way than balancing the number of cells. Any connected function is expected to take an iterator to a cell, and a CellStatus argument that indicates whether this cell is going to be refined, coarsened or left untouched (see the documentation of the CellStatus enum for more information). The function is expected to return an unsigned integer, which is interpreted as the additional computational load of this cell. If this cell is going to be coarsened, the signal is called for the parent cell and you need to provide the weight of the future parent cell. If this cell is going to be refined the function should return a weight, which will be equally assigned to every future child cell of the current cell. As a reference a value of 1000 is added for every cell to the total weight. This means a signal return value of 1000 (resulting in a weight of 2000) means that it is twice as expensive for a process to handle this particular cell. If several functions are connected to this signal, their return values will be summed to calculate the final weight.

This function is used in step-68.

Definition at line 2200 of file tria.h.

## ◆ pre_distributed_refinement

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_distributed_refinement

This signal is triggered at the beginning of execution of the parallel::distributed::Triangulation::execute_coarsening_and_refinement() function (which is itself called by other functions such as Triangulation::refine_global() ). At the time this signal is triggered, the triangulation is still unchanged. This signal is different from the pre_refinement signal, because in the parallel distributed case the pre_refinement signal is triggered multiple times without a way to distinguish the last signal call.

Definition at line 2213 of file tria.h.

## ◆ post_distributed_refinement

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::post_distributed_refinement

This signal is triggered at the end of execution of the parallel::distributed::Triangulation::execute_coarsening_and_refinement() function when the triangulation has reached its final state. This signal is different from the post_refinement signal, because in the parallel distributed case the post_refinement signal is triggered multiple times without a way to distinguish the last signal call.

Definition at line 2223 of file tria.h.

## ◆ pre_distributed_repartition

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_distributed_repartition

This signal is triggered at the beginning of execution of the parallel::distributed::Triangulation::repartition() function. At the time this signal is triggered, the triangulation is still unchanged.

Note
The parallel::distributed::Triangulation::repartition() function is also called by parallel::distributed::Triangulation::load(). Thus, the pre_distributed_repartition signal will be triggered after the pre_distributed_load one.

Definition at line 2235 of file tria.h.

## ◆ post_distributed_repartition

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::post_distributed_repartition

This signal is triggered at the end of execution of the parallel::distributed::Triangulation::repartition() function when the triangulation has reached its final state.

Definition at line 2242 of file tria.h.

## ◆ pre_distributed_save

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_distributed_save

This signal is triggered at the beginning of execution of the parallel::distributed::Triangulation::save() function. At the time this signal is triggered, the triangulation is still unchanged.

Definition at line 2250 of file tria.h.

## ◆ post_distributed_save

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::post_distributed_save

This signal is triggered at the end of execution of the parallel::distributed::Triangulation::save() function when the triangulation has reached its final state.

Definition at line 2257 of file tria.h.

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::pre_distributed_load

This signal is triggered at the beginning of execution of the parallel::distributed::Triangulation::load() function. At the time this signal is triggered, the triangulation is still unchanged.

Definition at line 2265 of file tria.h.

template<int dim, int spacedim = dim>
 boost::signals2::signal Triangulation< dim, spacedim >::Signals::post_distributed_load

This signal is triggered at the end of execution of the parallel::distributed::Triangulation::load() function when the triangulation has reached its final state.

Definition at line 2272 of file tria.h.

The documentation for this struct was generated from the following file: