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Angular Analogue of Linear Momentum

A metal rod o...

Question

A metal rod of length 'L' and mass " $m$ is pivoted at one end. A thin disk of mass ' $M$ and radius ' $R^{'} (< L)$ is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case $B)$ the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

Restoring torque in case

A =

Restoring torque in case

B

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Restoring torque in case

A <

Restoring torque in case

B

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Angular frequency for case

A >

Angular frequency for case

B

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Angular frequency for case

A <

Angular frequency for case

B

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Solution

The correct options are
A Restoring torque in case $A =$ Restoring torque in case $B$
C Angular frequency for case $A >$ Angular frequency for case $B$

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Similar questions

Q. A metaI rod of length '

L

' and mass '

m

' is pivoted at one end. A thin disk of mass '

M

' and radius

^{'} R^{'} (< L)

is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

Q. A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius

R (< L)

is attached at its centre to the free end. A thin disk of mass M and radius

R (< L)

is attached at its centre to the free end of the rod. Consider two ways the disc is attached: (case A). The disc is not free to rotate about its centre and (case B) the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

As shown below, a uniform disc is mounted to a fixed axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disc so that it will rotate with the disc. A block is attached to the rod.

Disc : mass = 3m, radius = R, moment of inertia about center $I_{D} = \frac{3}{2} m R^{2}$

Rod : mass = m, length = 2R, moment of inertia about one end $I_{R} = \frac{4}{3} m R^{2}$

Block : mass 2m

The system is held in equilibrium with the rod at an angle $θ_{0}$ to the vertical, as shown above, by a horizontal string of negligible mass one end attached to the disc and the other to a wall.

The tension in the string is

Q. A thin rod of mass

M

and length

a

is free to rotate in horizontal plane about a fixed vertical axis

O O^{'}

. A thin circular disc of mass

M

and of radius

a / 4

is pivoted on this rod with its center at a distance

a / 4

from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity

ω

and the disc rotating about its vertical axis with angular velocity

4 ω

. The total angular momentum of the system about the axis

O O^{'}

(\frac{M a^{2} ω}{48}) n

. The value of

n

Disc : mass = 3m, radius = R, moment of inertia about center $I_{D} = \frac{3}{2} m R^{2}$

Rod : mass = m, length = 2R, moment of inertia about one end $I_{R} = \frac{4}{3} m R^{2}$

Block : mass 2m

The system is held in equilibrium with the rod at an angle $θ_{0}$ to the vertical, as shown above, by a horizontal string of negligible mass one end attached to the disc and the other to a wall.

The string is now cut, and the disc-rod-block system is free to rotate. As disc rotates, the rod passes the horizontal position shown below. The linear speed of the end of the rod for the instant the rod is in the horizontal position is : (The arrow represent the direction of vector quantities)

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The correct options are A Restoring torque in case A= Restoring torque in case B C Angular frequency for case A> Angular frequency for case B

The correct options are
A Restoring torque in case $A =$ Restoring torque in case $B$
C Angular frequency for case $A >$ Angular frequency for case $B$