Equilibrium of rigid bodies and equations of rotational motion — practice questions

Question 1 · difficulty L2 · understanding

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle $\theta_1$ and in the next 2 s it rotates through an angle $\theta_2$. The ratio $\frac{\theta_2}{\theta_1}$ is $\_\_\_\_$ .

• A. 6
• B. 3
• C. 4
• D. ${\frac{1}{3}}$

Question 2 · difficulty L3 · understanding

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm . It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are $\_\_\_\_$ and $\_\_\_\_$ respectively.

• A. $0.128 \pi \mathrm{Nm}, 100$
• B. $0.0128 \pi \mathrm{Nm}, 50$
• C. $0.128 \pi \mathrm{Nm}, 50$
• D. $0.0128 \pi \mathrm{Nm}, 100$

Question 3 · difficulty L3 · understanding

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure. If the mass of each disc is 600 gm and applied torque between two discs is $43 \times 10^5$ dyne.cm, the angular acceleration of the discs about the given axis $A B$ is $\_\_\_\_$ $\mathrm{rad} / \mathrm{s}^2$.

• A. 100
• B. 22
• C. 27
• D. 11

Question 4 · difficulty L3 · understanding

A thin uniform rod $(X)$ of mass $M$ and length $L$ is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is $\_\_\_\_$ . ( $\mathrm{g}=$ gravitational acceleration)

• A. $\sqrt{\frac{3}{2} \frac{g}{L}}$
• B. $\sqrt{\frac{3 g}{L}}$
• C. $\frac{3}{\sqrt{2}} \sqrt{\frac{g}{L}}$
• D. $\frac{1}{\sqrt{2}} \sqrt{\frac{g}{L}}$

Question 5 · difficulty L3 · understanding

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M . Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is ________ (assume no slipping of string on pulley).

• A. $$\dfrac{(M-m) g}{\left[\left(\frac{8}{3}\right) M+m\right]}$$
• B. $\dfrac{(M - m)g}{2M + m}$
• C. $\dfrac{(M - m)g}{M + m}$
• D. $$\dfrac{(M-m) g}{\left[\left(\frac{13}{6}\right) M+m\right]}$$

Question 6 · difficulty L3 · understanding

A uniform rod of mass $m$ and length $l$ suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is $\_\_\_\_$ . $(g$ acceleration due to gravity)

• A. $\mathrm{mg} / \mathrm{s}$
• B. $m g / 4$
• C. $m g$
• D. $m g / 2$

Question 7 · difficulty L3 · understanding

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be:

• A. 20 rad/s
• B. 30 rad/s
• C. 10 rad/s
• D. 0 rad/s

Question 8 · difficulty L3 · understanding

A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by 3 m is ________ J. ($g = 10~\mathrm{m/s^2}$)

Question 9 · difficulty L4 · understanding

A cylindrical tube $A B$ of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $A B$ and passing through the edge at end $A$, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube)

• A. ${P}_{{B}}={P}_{{A}} {e}^{\left(\frac{M \omega^2 {l}^2}{3 R T}\right)}$
• B. $P_B=P_A$
• C. ${P}_{{B}}={P}_{{A}} {e}^{\left(\frac{{M} \omega^2 {l}^2}{2 {R T}}\right)}$
• D. $P_B=P_A e^{\left(\frac{M \omega^2 l^2}{R T}\right)}$