System of Particles and Rotational Motion MCQs with Answers

Ans: Option (c) Moment of inertia 

Explanation: A rigid body's rotational inertia is referred to as its moment of inertia. It is a quantity that determines the torque required to achieve a desired angular acceleration along a rotating axis, like how mass influences the force required to achieve the desired acceleration. 

Ans: Option (c) Axis of rotation 

Explanation: When a body rotates along an axis that passes through its center of mass, its angular momentum is directed along its axis of rotation.

Ans: Option (d) Different 

Explanation: The linear velocities of all the particles in the body differ in rotational motion. 

Let the body rotate about point X with angular velocity v. 

VA =r1 v 

VB =r2 v 

VC =r3 v 

As a result, the linear velocities of all particles in the body will differ. 

Ans: Option (d) None of the options 

Explanation: The position of the center of mass is determined by (1) its shape and (2) how mass is distributed on its shape. These two factors determine whether the center of gravity is within or outside the body. 

If a solid body has a regular structure and its mass is distributed uniformly throughout its surface (i.e., for symmetrical objects), its center of gravity must be located inside the body. 

However, if a solid body has an uneven structure and its mass is not distributed uniformly, the center of mass may be within or outside the body. 

Ans: Option (b) Constant velocity 

Explanation: For an isolated system, no external forces are acting and the internal forces cancel out in pairs. 

Therefore, the total force, F = 0 

M d(V)/dt = F = 0

=> d(V)/dt = 0 (as M, which is the mass cannot be 0) 

Therefore, V = constant

This shows us that the center of mass of an isolated system has a constant velocity.

Ans: Option (c) K 

Explanation: The radius of gyration can be defined as the imaginary distance from the centroid at which the area of cross-section is thought to be focused at a point to obtain the same moment of inertia. It is represented by the letter k. 

Ans:  Option (b) 

Explanation: The rotational kinetic energy is the kinetic energy of rotation of a rotating rigid body or system of particles, and is given by KE= 1/2 Iω², where I is the moment of inertia, or “rotational mass” of the rigid body or system of particles. 

Ans:  Option (d) Rolling motion 

Explanation: The rolling motion is a combination of both translational and rotational motion. The translational motion of a body is the motion of its center of mass. During a body's rolling motion, the surfaces in touch are briefly distorted. A limited region of both entities comes into touch with each other as a result of this deformation. This phenomenon has the overall effect of opposing motion through the component of the contact force parallel to the surface, resulting in friction. 

Ans: Option (b) False 

Explanation: The position of the center of mass is determined by the shape, size, and distribution of the body's mass. These factors determine whether the center of gravity is within or outside the body.

Ans: Option (a) 

Explanation: Given: Moment of inertia (I) = 3 kg-m²

Angular Velocity = 2 rad/s 

Mass (m) = 12kg 

To find: Velocity (v) 

= ½ Iω² = ½ mv²

= ½ A3A(2)² = ½ A12Av²

= ½ A6 = ½ A6v²

= v² = 6/6

Velocity (v) = 1 m/s

Ans: Option (b) 

Explanation: We know that, 

The kinetic energy in the rotation is K=½ Iω²

Angular momentum = L=Iω 

∴ K=½ Iω² = ½ (I×ω)×ω 

⇒K= ½ Lω 

⇒L= 2K/ω 

So, L2/L1 = K2/K1 × ω1/ω2 = ½ × ½

µ2ω1=ω2 and 2K2=K1 (Given) 

⇒L2/L1 =1/4 

L2=L/4 

Thus, Angular momentum becomes one-fourth of its initial value. 

Ans: Option (c) 

Explanation: For the ring,

 I=MR²

Acceleration of a body rolling on the inclined plane, 

A= (g sinα)/(1+(I/MR² ) ) 

A= (g sin30°)/(1+(MR²/MR² ) )............. (α=30°) 

A= g/4 

Ans: Option (d) 

Explanation: The greater the mass of the body, the greater the moment of inertia. The moment of inertia of a body varies when its axis of rotation changes. Objects of various forms and sizes have distinct moments of inertia. The moment of inertia of a body is determined by its mass, its axis of rotation, and its shape and size. 

Ans: Option (c) 

Explanation: Given, that the M.I. of a thin uniform ring centered on an axis perpendicular to the plane is 4kgm² = mr². 

Using the Perpendicular axis theorem, 

M.I. of a thin uniform ring about an axis passing through the center and in the plane of the ring is 2kgm² = 1/2 mr². 

Now, Using the Parallel axis theorem, 

M.I. of ring about an axis tangent to the ring and in a plane of the ring

1/2 mr² + md² = 3/2 mr²

= (3/2) 4 kgm²

= 6 kgm²

Ans: Option (d) 

Explanation: The angular momentum is mathematically written as

L = I x w

We know that angular velocity acts along the axis of rotation, then according to the above equation, the angular momentum also acts along the axis of rotation.

Ans: Option (b) 

Explanation: µI=2MR²

For solid cylinder 

I = (20× (0.5) × (0.5))/2

I=5 kgm²

Ans: Option (d) 

Explanation: L=m(r×v) 

The direction of (r×v), hence the direction of angular momentum remains the same. 

Ans: Option (b)  

Explanation: Let M and R be the disc's mass and radius, respectively. 

I is the moment of inertia concerning AB. 

Io = ½ MR² the moment of inertia around an axis passing through O and perpendicular to the plane. 

Ic = Io + MR² is the moment of inertia around an axis passing through C and perpendicular to the plane. 

Ic =3 Io 

Io = 2I using the perpendicular axis theorem 

As a result, Ic = 6I

Ans: Option (a) 

Explanation: Moment of inertia = Torque × Angular acceleration 

Torque τ=Iα 

Since α is the angular acceleration about which the torque is computed, where I is the moment of inertia of the body. 

Statement b and d are accurate, but Statement a is incorrect. 

As a result, assertions b and d are correct. 

F = ma (force equals mass times acceleration) 

As a result, assertion c is correct. 

Ans: Option (b) 

Explanation: Angular momentum will remain the same since external torque is zero. 

MI will increase since r increases (I=2/5 mr²) 

Angular velocity decreases since L=I x ω is conserved.

Rotational KE: K (rotational)=L² /2I decreases since I increase.  

Ans: Option (a) 

Explanation: Acceleration of the body (pure) rolling on the inclined plane,

a= (g sinα)/(1+(I/MR² ) ) 

Now as I(solid cylinder) < I(hollow cylinder) for same mass and radius 

a(solid cylinder) > a(hollow cylinder)

Thus, the solid cylinder will reach the bottom first. 

Ans: Option (a) 

Explanation: Definition of torque =Product of Moment of Inertia × Ratio of change of ω 

∴t=Iα 

Ans: Option (d) 

Explanation: A circular disc will have 2 times the mass of the semicircular disc. 

Moment of inertia of circular disc of mass 2M is 

Icirculardisc =½ (2M) R² = MR²

The moment of inertia of a semi-circular disc is half of the Moment of inertia of a circular disc due to symmetry. 

∴I(semicirculardisc) = ½ I(circulardisc) =½ MR²

Ans: Option (b) 

Explanation: 

Total distance covered by the particle

d=v1 t1 +v2 t2 +v3 t3 

µt1 =t2 =t3 =20sec 

d=20(3+4+5) = 240 m 

Total time, T= 3×20 =60sec. 

Avg velocity

v=d/t

=240/60

 =4 m/s 

Ans: Option (d) 

Explanation: 

Let initial energy be K and angular velocity be w after change become K’ and w’ 

ω′ =1.1ω(given) 

K=½ Iω²

K’=½ I(ω′)² 

=½ I (1.1ω)²

=1.21(½ Iω²) 

=1.21K 

Therefore, kinetic energy should be increased by 21%