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### IIT-JEE 2011 Paper 1 Offline

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

A
B
C
D

## Explanation

Due to upthrust, the spring will be compressed. Due to damping by the liquid, the final position will be smaller than the initial position. Hence choices (c) and (d) are not possible. Due to buoyancy, the block will move upwards. Hence, according to the given sign convention, position (x) is positive initially. When the system is released, x will decrease and momentum (p) will increase becoming maximum when the system reaches the mean position (x = 0) after which the momentum will decrease to zero when the oscillator reaches the extreme position, after which the momentum becomes negative. Hence the correct graph is (b).

2

### IIT-JEE 2011 Paper 1 Offline

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

The phase space diagram for simple harmonic motion is a circle centred at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E1 and E2 are the total mechanical energies respectively. Then

A
E1 = $$\sqrt2$$E2
B
E1 = 2E2
C
E1 = 4E2
D
E1 = 16E2

## Explanation

Energy of simple harmonic oscillator is

$$E = {1 \over 2}k{A^2}$$

where k is the force constant and A the amplitude of the oscillator. Since the oscillator is the same, the value of k is the same. Hence

$${E_1} = {1 \over 2}kA_1^2$$ and $${E_2} = {1 \over 2}kA_2^2$$

$$\therefore$$ $${{{E_1}} \over {{E_2}}} = {\left( {{{{A_1}} \over {{A_2}}}} \right)^2}$$

Now, A1 = maximum value of displacement of oscillator having energy E1 = 2a and A2 = a. Therefore

$${{{E_1}} \over {{E_2}}} = {\left( {{{2a} \over a}} \right)^2} = 4$$. So, $${E_1} = 4{E_2}$$

3

### IIT-JEE 2011 Paper 1 Offline

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

The phase space diagram for a ball thrown vertically up from ground is

A
B
C
D

## Explanation

Let the ball of mass m be thrown up with an initial velocity u. Its velocity v and displacement x are related by v2 $$-$$ u2 = $$-$$2gx, where g is the acceleration due to gravity. The momentum (p = mv) is given by

p2 = m2u2 $$-$$ 2m2gx,

which gives

$$p = \pm \sqrt {{m^2}{u^2} - 2{m^2}gx}$$.

At x = 0, the momentum is mu when the ball starts going up and it becomes $$-$$mu when the ball comes back. At the maximum height, x = u2/(2g), the momentum becomes zero.

4

### IIT-JEE 2011 Paper 2 Offline

A wooden block performs $$SHM$$ on a frictionless surface with frequency, $${v_0}.$$ The block carries a charge $$+Q$$ on its surface . If now a uniform electric field $$\overrightarrow E$$ is switched- on as shown, then the $$SHM$$ of the block will be
A
of the same frequency and with shifted mean position.
B
of the same frequency and with the same mean position
C
of changed frequency and with shifted mean position.
D
of changed frequency and with the same mean position.

## Explanation

The force exerted on charge +Q by the electric field $$\overrightarrow E$$ is

$$\overrightarrow F = Q\overrightarrow E$$

in the direction of $$\overrightarrow E$$. Since $$\overrightarrow F$$ is constant, a constant force is added to the applied force. Hence only the mean position will change.

The frequency will be same.

As $${v_0} = {1 \over {2\pi }}\sqrt {{k \over m}}$$ does not depend on the constant external force.

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