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PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365637 A body of mass $m$ and radius $r$ is released from rest along a smooth inclined plane of angle of inclination $θ$ . The angular momentum of the body about the instantaneous point of contact after a time $t$ from the instant of release is equal to

1

m g r t \cos θ

2

(\frac{3}{2}) m g r t \sin θ

3

m g r t \sin θ

4 None of these

Explanation:

Since surface is frictionless, the body does not rotate about its centre of mass. Only it slides parallel to the surface, i.e., $ω = 0$ and $v = (g \sin θ) t$
The angular momentum after time $t$ :
$L = m v r = m (g \sin θ t) r = m g r t \sin θ$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365638 A disc of mass $M$ and radius $R$ is rolling with angular speed $ω$ on a horizontal plane as shown in the figure. The magnitude of angular momentum of the disc about the origin $O$ is
supporting img

1

2 M R^{2} ω

2

\frac{1}{2} M R^{2} ω

3

\frac{3}{2} M R^{2} ω

4

M R^{2} ω

Explanation:

Angular momentum about
$\begin{aligned} origin = I_{translation} + I_{rotation} \\ = M v R + I_{c} ω = M (R ω) R + \frac{1}{2} M R^{2} ω = \frac{3}{2} M R^{2} ω \end{aligned}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365639 Assertion :
For the planets orbiting around the sun, angular speed, linear speed and K.E. change with time, but angular momentum remains constant.
Reason :
No torque is acting on the rotating planet. So its angular momentum is constant.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365640 A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$ . The child now stretches his arm so that moment of inerita of the system doubles. The kinetic energy of the system now is:

1

2 K

2

\frac{K}{2}

3

\frac{K}{4}

4

4 K

Explanation:

$L = I ω =$ constant
Kinetic energy $K = \frac{L^{2}}{2 I} \Rightarrow K α \frac{1}{I}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365637 A body of mass $m$ and radius $r$ is released from rest along a smooth inclined plane of angle of inclination $θ$ . The angular momentum of the body about the instantaneous point of contact after a time $t$ from the instant of release is equal to

1

m g r t \cos θ

2

(\frac{3}{2}) m g r t \sin θ

3

m g r t \sin θ

4 None of these

Explanation:

Since surface is frictionless, the body does not rotate about its centre of mass. Only it slides parallel to the surface, i.e., $ω = 0$ and $v = (g \sin θ) t$
The angular momentum after time $t$ :
$L = m v r = m (g \sin θ t) r = m g r t \sin θ$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365638 A disc of mass $M$ and radius $R$ is rolling with angular speed $ω$ on a horizontal plane as shown in the figure. The magnitude of angular momentum of the disc about the origin $O$ is
supporting img

1

2 M R^{2} ω

2

\frac{1}{2} M R^{2} ω

3

\frac{3}{2} M R^{2} ω

4

M R^{2} ω

Explanation:

Angular momentum about
$\begin{aligned} origin = I_{translation} + I_{rotation} \\ = M v R + I_{c} ω = M (R ω) R + \frac{1}{2} M R^{2} ω = \frac{3}{2} M R^{2} ω \end{aligned}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365639 Assertion :
For the planets orbiting around the sun, angular speed, linear speed and K.E. change with time, but angular momentum remains constant.
Reason :
No torque is acting on the rotating planet. So its angular momentum is constant.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365640 A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$ . The child now stretches his arm so that moment of inerita of the system doubles. The kinetic energy of the system now is:

1

2 K

2

\frac{K}{2}

3

\frac{K}{4}

4

4 K

Explanation:

$L = I ω =$ constant
Kinetic energy $K = \frac{L^{2}}{2 I} \Rightarrow K α \frac{1}{I}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365637 A body of mass $m$ and radius $r$ is released from rest along a smooth inclined plane of angle of inclination $θ$ . The angular momentum of the body about the instantaneous point of contact after a time $t$ from the instant of release is equal to

1

m g r t \cos θ

2

(\frac{3}{2}) m g r t \sin θ

3

m g r t \sin θ

4 None of these

Explanation:

Since surface is frictionless, the body does not rotate about its centre of mass. Only it slides parallel to the surface, i.e., $ω = 0$ and $v = (g \sin θ) t$
The angular momentum after time $t$ :
$L = m v r = m (g \sin θ t) r = m g r t \sin θ$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365638 A disc of mass $M$ and radius $R$ is rolling with angular speed $ω$ on a horizontal plane as shown in the figure. The magnitude of angular momentum of the disc about the origin $O$ is
supporting img

1

2 M R^{2} ω

2

\frac{1}{2} M R^{2} ω

3

\frac{3}{2} M R^{2} ω

4

M R^{2} ω

Explanation:

Angular momentum about
$\begin{aligned} origin = I_{translation} + I_{rotation} \\ = M v R + I_{c} ω = M (R ω) R + \frac{1}{2} M R^{2} ω = \frac{3}{2} M R^{2} ω \end{aligned}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365639 Assertion :
For the planets orbiting around the sun, angular speed, linear speed and K.E. change with time, but angular momentum remains constant.
Reason :
No torque is acting on the rotating planet. So its angular momentum is constant.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365640 A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$ . The child now stretches his arm so that moment of inerita of the system doubles. The kinetic energy of the system now is:

1

2 K

2

\frac{K}{2}

3

\frac{K}{4}

4

4 K

Explanation:

$L = I ω =$ constant
Kinetic energy $K = \frac{L^{2}}{2 I} \Rightarrow K α \frac{1}{I}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365637 A body of mass $m$ and radius $r$ is released from rest along a smooth inclined plane of angle of inclination $θ$ . The angular momentum of the body about the instantaneous point of contact after a time $t$ from the instant of release is equal to

1

m g r t \cos θ

2

(\frac{3}{2}) m g r t \sin θ

3

m g r t \sin θ

4 None of these

Explanation:

Since surface is frictionless, the body does not rotate about its centre of mass. Only it slides parallel to the surface, i.e., $ω = 0$ and $v = (g \sin θ) t$
The angular momentum after time $t$ :
$L = m v r = m (g \sin θ t) r = m g r t \sin θ$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365638 A disc of mass $M$ and radius $R$ is rolling with angular speed $ω$ on a horizontal plane as shown in the figure. The magnitude of angular momentum of the disc about the origin $O$ is
supporting img

1

2 M R^{2} ω

2

\frac{1}{2} M R^{2} ω

3

\frac{3}{2} M R^{2} ω

4

M R^{2} ω

Explanation:

Angular momentum about
$\begin{aligned} origin = I_{translation} + I_{rotation} \\ = M v R + I_{c} ω = M (R ω) R + \frac{1}{2} M R^{2} ω = \frac{3}{2} M R^{2} ω \end{aligned}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365639 Assertion :
For the planets orbiting around the sun, angular speed, linear speed and K.E. change with time, but angular momentum remains constant.
Reason :
No torque is acting on the rotating planet. So its angular momentum is constant.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365640 A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$ . The child now stretches his arm so that moment of inerita of the system doubles. The kinetic energy of the system now is:

1

2 K

2

\frac{K}{2}

3

\frac{K}{4}

4

4 K

Explanation:

$L = I ω =$ constant
Kinetic energy $K = \frac{L^{2}}{2 I} \Rightarrow K α \frac{1}{I}$