Grade 11 : Physics - Rotational Motion (Set 3)

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Grade 11 : Physics - Rotational Motion (Set 3)— Questions & Detailed Solutions

Two point masses $m_1 = 2kg$ and $m_2 = 3kg$ are connected by a rigid, light rod of length $L = 10m$.
What is the distance of the center of mass of the system measured from the 2 kg mass?

(A)

4 m

(B)

6 m

(C)

5 m

(D)

3 m

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Placing $m_1$ at the origin ($x_1 = 0$), the position of mass $m_2$ is $x_2 = 10m$. The position of the center of mass is given by: $$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{(2)(0) + (3)(10)}{2 + 3} = \frac{30}{5} = 6\text{ m}$$ Thus, the center of mass is 6 m away from the 2 kg mass.

A thin uniform rod has a total mass $M = 4kg$ and length $L = 3m$.
Calculate its moment of inertia about an axis perpendicular to the rod and passing through one of its ends.

(A)

$36kg\cdotm^2$

(B)

$4kg\cdotm^2$

(C)

$12kg\cdotm^2$

(D)

$18kg\cdotm^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: The moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis through one end is: $$I_{end} = \frac{1}{3} M L^2$$ Substituting $M = 4kg$ and $L = 3m$: $$I_{end} = \frac{1}{3} \times 4 \times (3)^2 = \frac{1}{3} \times 4 \times 9 = 12\text{ kg}\cdot\text{m}^2$$

A solid sphere of mass $M = 5kg$ and radius $R = 2m$ is considered along with a line tangent to its outer surface.
Using the parallel axis theorem, find the moment of inertia of the solid sphere about an axis tangent to its surface.

(A)

$28kg\cdotm^2$

(B)

$20kg\cdotm^2$

(C)

$14kg\cdotm^2$

(D)

$8kg\cdotm^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: For a solid sphere about a central axis, $I_{cm} = \frac{2}{5} M R^2$. By the parallel axis theorem: $$I_{tangent} = I_{cm} + M R^2 = \frac{7}{5} M R^2$$ Substituting $M = 5kg$ and $R = 2m$: $$I_{tangent} = \frac{7}{5} (5) (2)^2 = 7 \times 4 = 28\text{ kg}\cdot\text{m}^2$$

A heavy flywheel starts rotating from rest under a constant angular acceleration $\alpha = 4\text{ rad/s}^2$.
What is the magnitude of its angular velocity after $t = 5seconds$?

(A)

10 rad/s

(B)

40 rad/s

(C)

50 rad/s

(D)

20 rad/s

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation: Using the rotational kinematic equation $\omega = \omega_0 + \alpha t$ with initial angular velocity $\omega_0 = 0\text{ rad/s}$: $$\omega = 0 + (4\text{ rad/s}^2)(5\text{ s}) = 20\text{ rad/s}$$

A particle of mass $m = 0.5kg$ moves along a straight line $y = 3m$ with a constant velocity of $v = 4m/s$ parallel to the x-axis.
What is the magnitude of the angular momentum of the particle about the origin?

(A)

$12kg\cdotm^2/\text{s}$

(B)

$6kg\cdotm^2/\text{s}$

(C)

$3kg\cdotm^2/\text{s}$

(D)

$2.5kg\cdotm^2/\text{s}$

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Angular momentum is given by $L = m v r_{\perp}$, where $r_{\perp}$ is the perpendicular distance from the origin to the line of motion. Here $r_{\perp} = 3m$. $$L = (0.5\text{ kg}) \times (4\text{ m/s}) \times (3\text{ m}) = 6\text{ kg}\cdot\text{m}^2/\text{s}$$

A turntable of moment of inertia $I_1 = 400kg\cdotm^2$ rotates freely about a vertical axis at an angular speed $\omega_1 = 3\text{ rad/s}$. A child steps onto the outer rim, increasing the total moment of inertia to $I_2 = 600kg\cdotm^2$.
Assuming no external torque acts on the system, what is the new angular velocity of the turntable?

(A)

2 rad/s

(B)

4.5 rad/s

(C)

1.5 rad/s

(D)

3 rad/s

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: By conservation of angular momentum ($L_i = L_f$): $$I_1 \omega_1 = I_2 \omega_2$$ $$(400)(3) = (600) \omega_2 \implies 1200 = 600 \omega_2 \implies \omega_2 = 2\text{ rad/s}$$

A net torque $\tau = 50\text{ N}\cdotm$ is applied to a rigid body having a moment of inertia $I = 10kg\cdotm^2$ about its rotation axis.
Find the angular acceleration produced in the body.

(A)

$500\text{ rad/s}^2$

(B)

$0.2\text{ rad/s}^2$

(C)

$5\text{ rad/s}^2$

(D)

$25\text{ rad/s}^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: Rotational analog of Newton's second law is $\tau = I \alpha$. $$\alpha = \frac{\tau}{I} = \frac{50\text{ N}\cdot\text{m}}{10\text{ kg}\cdot\text{m}^2} = 5\text{ rad/s}^2$$

A uniform solid disc has mass $M = 2kg$ and radius $R = 0.5m$. It rotates about an axis passing through its center perpendicular to its plane at $\omega = 10\text{ rad/s}$.
Determine the rotational kinetic energy of the disc.

(A)

25 J

(B)

12.5 J

(C)

50 J

(D)

6.25 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Moment of inertia $I = \frac{1}{2} M R^2 = \frac{1}{2} (2) (0.5)^2 = 0.25kg\cdotm^2$. Rotational kinetic energy: $$K_{rot} = \frac{1}{2} I \omega^2 = \frac{1}{2} (0.25) (10)^2 = 0.125 \times 100 = 12.5\text{ J}$$

A solid cylinder rolls without slipping on a horizontal plane.
What is the ratio of its rotational kinetic energy to its total kinetic energy?

(A)

$1/2$

(B)

$2/3$

(C)

$1/4$

(D)

$1/3$

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation: For pure rolling, $v = R \omega$. $K_{rot} = \frac{1}{2} I \omega^2 = \frac{1}{2} (\frac{1}{2} M R^2) (v^2/R^2) = \frac{1}{4} M v^2$. $K_{trans} = \frac{1}{2} M v^2$. Thus, $K_{total} = K_{trans} + K_{rot} = \frac{3}{4} M v^2$. $$ ext{Ratio} = \frac{K_{rot}}{K_{total}} = \frac{\frac{1}{4} M v^2}{\frac{3}{4} M v^2} = \frac{1}{3}$$

Q10

Consider a thin uniform circular ring of radius $R = 6cm$.
Calculate the radius of gyration of the ring about an axis lying along one of its diameters.

(A)

$3\sqrt{2}cm$

(B)

3 cm

(C)

6 cm

(D)

4.5 cm

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: For a ring about its diameter, $I_{dia} = \frac{1}{2} M R^2$. Equating $I = M k^2$, where $k$ is the radius of gyration: $$M k^2 = \frac{1}{2} M R^2 \implies k = \frac{R}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2}\text{ cm}$$

Q11

Three particles of masses 1 kg, 2 kg, and 3 kg are located at the vertices $(0,0)$, $(2,0)$, and $(1, \sqrt{3})$ of an equilateral triangle in meters.
Find the coordinates of the center of mass of this three-particle system.

(A)

$(1, \sqrt{3}/3)$

(B)

$(7/6, \sqrt{3})$

(C)

$(7/6, \sqrt{3}/2)$

(D)

$(5/6, \sqrt{3}/2)$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: Total mass $M = 1 + 2 + 3 = 6kg$. $$x_{cm} = \frac{1(0) + 2(2) + 3(1)}{6} = \frac{7}{6}\text{ m}$$ $$y_{cm} = \frac{1(0) + 2(0) + 3(\sqrt{3})}{6} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}\text{ m}$$ Coordinates of COM are $(7/6, \sqrt{3}/2)$.

Q12

A heavy wheel of moment of inertia $I = 4kg\cdotm^2$ is rotating at an initial angular speed $\omega_0 = 6\text{ rad/s}$. A friction brake exerts a torque to bring it completely to rest.
What is the magnitude of work done by the retarding torque on the wheel?

(A)

144 J

(B)

72 J

(C)

36 J

(D)

24 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: By the Work-Energy Theorem for rotation, $W = \Delta K = K_f - K_i$. $$K_i = \frac{1}{2} I \omega_0^2 = \frac{1}{2} (4) (6^2) = 2 \times 36 = 72\text{ J}$$ Since $K_f = 0$, the magnitude of work done to stop the wheel is 72 J.

Q13

A uniform square plate has mass $M = 12kg$ and side length $a = 2m$.
Using the perpendicular axis theorem, calculate its moment of inertia about an axis passing through its center and lying in its plane parallel to one of its sides.

(A)

$4kg\cdotm^2$

(B)

$8kg\cdotm^2$

(C)

$12kg\cdotm^2$

(D)

$2kg\cdotm^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: For a square plate, $I_z$ (perpendicular to plane through center) is $\frac{1}{6} M a^2$. By symmetry, $I_x = I_y$ for axes in the plane. By perpendicular axis theorem, $I_z = I_x + I_y = 2 I_x$, so: $$I_x = \frac{1}{12} M a^2 = \frac{1}{12} (12) (2^2) = 4\text{ kg}\cdot\text{m}^2$$

Q14

A solid sphere of radius $R$ is allowed to roll down an inclined plane of inclination $\theta = 30^\circ$ without slipping. Acceleration due to gravity is $g = 9.8m/s^2$.
Calculate the linear acceleration of the center of mass of the sphere.

(A)

$4.9m/s^2$

(B)

$2.45m/s^2$

(C)

$7.0m/s^2$

(D)

$3.5m/s^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation: For a rolling sphere on an incline, $a = \frac{g \sin\theta}{1 + k^2/R^2}$. For a solid sphere, $k^2/R^2 = 2/5$. $$a = \frac{g \sin 30^\circ}{1 + 2/5} = \frac{5}{7} g (0.5) = \frac{5}{7} \times 4.9 = 3.5\text{ m/s}^2$$

Q15

A uniform ladder of length 5 m and mass 20 kg rests against a smooth vertical wall with its lower end on a rough horizontal floor at a distance of 3 m from the wall. Assume $g = 10m/s^2$.
What is the magnitude of the normal reaction force exerted by the wall on the upper end of the ladder?

(A)

100 N

(B)

150 N

(C)

75 N

(D)

200 N

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: Height on wall $h = \sqrt{5^2 - 3^2} = 4m$. Take torque about foot of ladder at rotational equilibrium: $$\tau_{foot} = N_{wall} \times 4 - W \times 1.5 = 0$$ $$4 N_{wall} = (20 \times 10) \times 1.5 = 300 \implies N_{wall} = 75\text{ N}$$

Q16

A solid sphere is released from rest from the top of an inclined plane of vertical height $h = 7m$ and rolls down without slipping ($g = 10m/s^2$).
What is the linear speed of its center of mass when it reaches the bottom of the incline?

(A)

11.8 m/s

(B)

10 m/s

(C)

8.36 m/s

(D)

14 m/s

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Conservation of mechanical energy for pure rolling: $$M g h = \frac{1}{2} M v^2 \left(1 + \frac{k^2}{R^2}\right)$$ For a solid sphere, $k^2/R^2 = 2/5$, so $1 + k^2/R^2 = 7/5$. $$g h = \frac{7}{10} v^2 \implies v = \sqrt{\frac{10 g h}{7}} = \sqrt{\frac{10 \times 10 \times 7}{7}} = \sqrt{100} = 10\text{ m/s}$$

Q17

A constant torque of magnitude $\tau = 20\text{ N}\cdotm$ acts on a rotating flywheel for a time interval of $t = 3seconds$.
Calculate the magnitude of change in angular momentum of the flywheel.

(A)

$60kg\cdotm^2/\text{s}$

(B)

$20kg\cdotm^2/\text{s}$

(C)

$6.67kg\cdotm^2/\text{s}$

(D)

$120kg\cdotm^2/\text{s}$

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Angular impulse equals the change in angular momentum: $$\Delta L = \tau \Delta t = (20\text{ N}\cdot\text{m}) \times (3\text{ s}) = 60\text{ kg}\cdot\text{m}^2/\text{s}$$

Q18

A uniform rod of mass $M = 6kg$ and length $L = 2m$ is tilted so that an axis passes through its center of mass making an angle of $\theta = 60^\circ$ with its length.
Find the moment of inertia of the rod about this axis.

(A)

$2.0kg\cdotm^2$

(B)

$0.75kg\cdotm^2$

(C)

$1.5kg\cdotm^2$

(D)

$3.0kg\cdotm^2$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: The moment of inertia of an inclined rod about a central axis at angle $\theta$ to its length is: $$I = \frac{1}{12} M L^2 \sin^2\theta$$ $$I = \frac{1}{12} (6) (2^2) \sin^2(60^\circ) = 2 \times \left(\frac{\sqrt{3}}{2}\right)^2 = 2 \times \frac{3}{4} = 1.5\text{ kg}\cdot\text{m}^2$$

Q19

A circular disc of moment of inertia $I_1 = 2kg\cdotm^2$ rotates horizontally about a vertical axis at $\omega_1 = 10\text{ rad/s}$. Another non-rotating disc of moment of inertia $I_2 = 3kg\cdotm^2$ is dropped concentrically onto it.
Calculate the total loss in kinetic energy due to friction between the two discs.

(A)

40 J

(B)

100 J

(C)

20 J

(D)

60 J

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation: By angular momentum conservation, final angular velocity $\omega_f = \frac{I_1 \omega_1}{I_1 + I_2} = \frac{2 \times 10}{5} = 4\text{ rad/s}$. $$K_i = \frac{1}{2} (2) (10)^2 = 100\text{ J}$$ $$K_f = \frac{1}{2} (5) (4)^2 = 40\text{ J}$$ $$\text{Loss in } K = K_i - K_f = 100 - 40 = 60\text{ J}$$

Q20

A wheel of radius $R = 0.4m$ rolls without slipping on a horizontal ground surface.
If the linear speed of the top-most point of the wheel relative to the ground is 12 m/s, what is the linear speed of its center of mass?

(A)

12 m/s

(B)

6 m/s

(C)

3 m/s

(D)

24 m/s

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: In pure rolling on a stationary surface, the top-most point has speed $v_{top} = v_{cm} + R \omega = 2 v_{cm}$. $$v_{cm} = \frac{v_{top}}{2} = \frac{12\text{ m/s}}{2} = 6\text{ m/s}$$

Grade 11 : Physics - Rotational Motion (Set 3)

Grade 11 : Physics - Rotational Motion (Set 3)— Questions & Detailed Solutions

Two point masses $m_1 = 2kg$ and $m_2 = 3kg$ are connected by a rigid, light rod of length $L = 10m$.
What is the distance of the center of mass of the system measured from the 2 kg mass?

A thin uniform rod has a total mass $M = 4kg$ and length $L = 3m$.
Calculate its moment of inertia about an axis perpendicular to the rod and passing through one of its ends.

A solid sphere of mass $M = 5kg$ and radius $R = 2m$ is considered along with a line tangent to its outer surface.
Using the parallel axis theorem, find the moment of inertia of the solid sphere about an axis tangent to its surface.

A heavy flywheel starts rotating from rest under a constant angular acceleration $\alpha = 4\text{ rad/s}^2$.
What is the magnitude of its angular velocity after $t = 5seconds$?

A particle of mass $m = 0.5kg$ moves along a straight line $y = 3m$ with a constant velocity of $v = 4m/s$ parallel to the x-axis.
What is the magnitude of the angular momentum of the particle about the origin?

A net torque $\tau = 50\text{ N}\cdotm$ is applied to a rigid body having a moment of inertia $I = 10kg\cdotm^2$ about its rotation axis.
Find the angular acceleration produced in the body.

A uniform solid disc has mass $M = 2kg$ and radius $R = 0.5m$. It rotates about an axis passing through its center perpendicular to its plane at $\omega = 10\text{ rad/s}$.
Determine the rotational kinetic energy of the disc.

A solid cylinder rolls without slipping on a horizontal plane.
What is the ratio of its rotational kinetic energy to its total kinetic energy?

Consider a thin uniform circular ring of radius $R = 6cm$.
Calculate the radius of gyration of the ring about an axis lying along one of its diameters.

Three particles of masses 1 kg, 2 kg, and 3 kg are located at the vertices $(0,0)$, $(2,0)$, and $(1, \sqrt{3})$ of an equilateral triangle in meters.
Find the coordinates of the center of mass of this three-particle system.

A heavy wheel of moment of inertia $I = 4kg\cdotm^2$ is rotating at an initial angular speed $\omega_0 = 6\text{ rad/s}$. A friction brake exerts a torque to bring it completely to rest.
What is the magnitude of work done by the retarding torque on the wheel?

A uniform square plate has mass $M = 12kg$ and side length $a = 2m$.
Using the perpendicular axis theorem, calculate its moment of inertia about an axis passing through its center and lying in its plane parallel to one of its sides.

A solid sphere of radius $R$ is allowed to roll down an inclined plane of inclination $\theta = 30^\circ$ without slipping. Acceleration due to gravity is $g = 9.8m/s^2$.
Calculate the linear acceleration of the center of mass of the sphere.

A uniform ladder of length 5 m and mass 20 kg rests against a smooth vertical wall with its lower end on a rough horizontal floor at a distance of 3 m from the wall. Assume $g = 10m/s^2$.
What is the magnitude of the normal reaction force exerted by the wall on the upper end of the ladder?

A solid sphere is released from rest from the top of an inclined plane of vertical height $h = 7m$ and rolls down without slipping ($g = 10m/s^2$).
What is the linear speed of its center of mass when it reaches the bottom of the incline?

A constant torque of magnitude $\tau = 20\text{ N}\cdotm$ acts on a rotating flywheel for a time interval of $t = 3seconds$.
Calculate the magnitude of change in angular momentum of the flywheel.

A uniform rod of mass $M = 6kg$ and length $L = 2m$ is tilted so that an axis passes through its center of mass making an angle of $\theta = 60^\circ$ with its length.
Find the moment of inertia of the rod about this axis.

A wheel of radius $R = 0.4m$ rolls without slipping on a horizontal ground surface.
If the linear speed of the top-most point of the wheel relative to the ground is 12 m/s, what is the linear speed of its center of mass?

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Grade 11 : Physics - Rotational Motion (Set 3)

Grade 11 : Physics - Rotational Motion (Set 3)— Questions & Detailed Solutions

Two point masses $m_1 = 2kg$ and $m_2 = 3kg$ are connected by a rigid, light rod of length $L = 10m$.What is the distance of the center of mass of the system measured from the 2 kg mass?

A thin uniform rod has a total mass $M = 4kg$ and length $L = 3m$.Calculate its moment of inertia about an axis perpendicular to the rod and passing through one of its ends.

A solid sphere of mass $M = 5kg$ and radius $R = 2m$ is considered along with a line tangent to its outer surface.Using the parallel axis theorem, find the moment of inertia of the solid sphere about an axis tangent to its surface.

A heavy flywheel starts rotating from rest under a constant angular acceleration $\alpha = 4\text{ rad/s}^2$.What is the magnitude of its angular velocity after $t = 5seconds$?

A particle of mass $m = 0.5kg$ moves along a straight line $y = 3m$ with a constant velocity of $v = 4m/s$ parallel to the x-axis.What is the magnitude of the angular momentum of the particle about the origin?

A net torque $\tau = 50\text{ N}\cdotm$ is applied to a rigid body having a moment of inertia $I = 10kg\cdotm^2$ about its rotation axis.Find the angular acceleration produced in the body.

A uniform solid disc has mass $M = 2kg$ and radius $R = 0.5m$. It rotates about an axis passing through its center perpendicular to its plane at $\omega = 10\text{ rad/s}$.Determine the rotational kinetic energy of the disc.

A solid cylinder rolls without slipping on a horizontal plane.What is the ratio of its rotational kinetic energy to its total kinetic energy?

Consider a thin uniform circular ring of radius $R = 6cm$.Calculate the radius of gyration of the ring about an axis lying along one of its diameters.

Three particles of masses 1 kg, 2 kg, and 3 kg are located at the vertices $(0,0)$, $(2,0)$, and $(1, \sqrt{3})$ of an equilateral triangle in meters.Find the coordinates of the center of mass of this three-particle system.

A heavy wheel of moment of inertia $I = 4kg\cdotm^2$ is rotating at an initial angular speed $\omega_0 = 6\text{ rad/s}$. A friction brake exerts a torque to bring it completely to rest.What is the magnitude of work done by the retarding torque on the wheel?

A uniform square plate has mass $M = 12kg$ and side length $a = 2m$.Using the perpendicular axis theorem, calculate its moment of inertia about an axis passing through its center and lying in its plane parallel to one of its sides.

A solid sphere of radius $R$ is allowed to roll down an inclined plane of inclination $\theta = 30^\circ$ without slipping. Acceleration due to gravity is $g = 9.8m/s^2$.Calculate the linear acceleration of the center of mass of the sphere.

A uniform ladder of length 5 m and mass 20 kg rests against a smooth vertical wall with its lower end on a rough horizontal floor at a distance of 3 m from the wall. Assume $g = 10m/s^2$.What is the magnitude of the normal reaction force exerted by the wall on the upper end of the ladder?

A solid sphere is released from rest from the top of an inclined plane of vertical height $h = 7m$ and rolls down without slipping ($g = 10m/s^2$).What is the linear speed of its center of mass when it reaches the bottom of the incline?

A constant torque of magnitude $\tau = 20\text{ N}\cdotm$ acts on a rotating flywheel for a time interval of $t = 3seconds$.Calculate the magnitude of change in angular momentum of the flywheel.

A uniform rod of mass $M = 6kg$ and length $L = 2m$ is tilted so that an axis passes through its center of mass making an angle of $\theta = 60^\circ$ with its length.Find the moment of inertia of the rod about this axis.

A wheel of radius $R = 0.4m$ rolls without slipping on a horizontal ground surface.If the linear speed of the top-most point of the wheel relative to the ground is 12 m/s, what is the linear speed of its center of mass?

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Two point masses $m_1 = 2kg$ and $m_2 = 3kg$ are connected by a rigid, light rod of length $L = 10m$.
What is the distance of the center of mass of the system measured from the 2 kg mass?

A thin uniform rod has a total mass $M = 4kg$ and length $L = 3m$.
Calculate its moment of inertia about an axis perpendicular to the rod and passing through one of its ends.

A solid sphere of mass $M = 5kg$ and radius $R = 2m$ is considered along with a line tangent to its outer surface.
Using the parallel axis theorem, find the moment of inertia of the solid sphere about an axis tangent to its surface.

A heavy flywheel starts rotating from rest under a constant angular acceleration $\alpha = 4\text{ rad/s}^2$.
What is the magnitude of its angular velocity after $t = 5seconds$?

A particle of mass $m = 0.5kg$ moves along a straight line $y = 3m$ with a constant velocity of $v = 4m/s$ parallel to the x-axis.
What is the magnitude of the angular momentum of the particle about the origin?

A net torque $\tau = 50\text{ N}\cdotm$ is applied to a rigid body having a moment of inertia $I = 10kg\cdotm^2$ about its rotation axis.
Find the angular acceleration produced in the body.

A uniform solid disc has mass $M = 2kg$ and radius $R = 0.5m$. It rotates about an axis passing through its center perpendicular to its plane at $\omega = 10\text{ rad/s}$.
Determine the rotational kinetic energy of the disc.

A solid cylinder rolls without slipping on a horizontal plane.
What is the ratio of its rotational kinetic energy to its total kinetic energy?

Consider a thin uniform circular ring of radius $R = 6cm$.
Calculate the radius of gyration of the ring about an axis lying along one of its diameters.

Three particles of masses 1 kg, 2 kg, and 3 kg are located at the vertices $(0,0)$, $(2,0)$, and $(1, \sqrt{3})$ of an equilateral triangle in meters.
Find the coordinates of the center of mass of this three-particle system.

A heavy wheel of moment of inertia $I = 4kg\cdotm^2$ is rotating at an initial angular speed $\omega_0 = 6\text{ rad/s}$. A friction brake exerts a torque to bring it completely to rest.
What is the magnitude of work done by the retarding torque on the wheel?

A uniform square plate has mass $M = 12kg$ and side length $a = 2m$.
Using the perpendicular axis theorem, calculate its moment of inertia about an axis passing through its center and lying in its plane parallel to one of its sides.

A solid sphere of radius $R$ is allowed to roll down an inclined plane of inclination $\theta = 30^\circ$ without slipping. Acceleration due to gravity is $g = 9.8m/s^2$.
Calculate the linear acceleration of the center of mass of the sphere.

A uniform ladder of length 5 m and mass 20 kg rests against a smooth vertical wall with its lower end on a rough horizontal floor at a distance of 3 m from the wall. Assume $g = 10m/s^2$.
What is the magnitude of the normal reaction force exerted by the wall on the upper end of the ladder?

A solid sphere is released from rest from the top of an inclined plane of vertical height $h = 7m$ and rolls down without slipping ($g = 10m/s^2$).
What is the linear speed of its center of mass when it reaches the bottom of the incline?

A constant torque of magnitude $\tau = 20\text{ N}\cdotm$ acts on a rotating flywheel for a time interval of $t = 3seconds$.
Calculate the magnitude of change in angular momentum of the flywheel.

A uniform rod of mass $M = 6kg$ and length $L = 2m$ is tilted so that an axis passes through its center of mass making an angle of $\theta = 60^\circ$ with its length.
Find the moment of inertia of the rod about this axis.

A wheel of radius $R = 0.4m$ rolls without slipping on a horizontal ground surface.
If the linear speed of the top-most point of the wheel relative to the ground is 12 m/s, what is the linear speed of its center of mass?