JEE Physics System Of Particles and Rotational Motion MCQs Set 63

Physics Objective Questions and Answers: System Of Particles and Rotational Motion

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Multiple Choice Questions

Question 1: When a mass is rotating in a plane about a fixed point, its angular momentum is directed along the
(a) radius of the orbit
(b) tangent to the orbit
(c) line parallel to the plane of rotation
(d) line perpendicular to the plane of rotation
Show Answer & Explanation

Answer: (d) line perpendicular to the plane of rotation

Explanation:
1. L⃗ = r⃗ × p⃗ is perpendicular to both r⃗ and p⃗.
2. Both lie in the plane of rotation, so L⃗ is perpendicular to that plane.

Teacher's Note:
1. The cross product is perpendicular to both vectors.

Question 2: A shell following a parabolic path explodes somewhere in its flight. The centre of mass of the fragments will continue to move in
(a) a vertical direction
(b) any direction
(c) a horizontal direction
(d) the same parabolic path
Show Answer & Explanation

Answer: (d) the same parabolic path

Explanation:
1. The explosion is caused by internal forces.
2. Only gravity acts externally, so the CM continues along the original parabola.

Teacher's Note:
1. Explosion forces are internal.

Question 3: A body rolls down an inclined plane. If its kinetic energy of rotation is 40% of its kinetic energy of translation, then the body is
(a) hollow cylinder
(b) ring
(c) solid disc
(d) solid sphere
Show Answer & Explanation

Answer: (d) solid sphere

Explanation:
1. K_rot/K_trans = I/(mR²) = k²/R² = 0.4.
2. The solid sphere has k²/R² = 2/5.

Teacher's Note:
1. Match the value of k²/R².

Question 4: If a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping, then
(a) ring has the least velocity of the centre of mass at the bottom of the inclined plane
(b) sphere has the least velocity of the centre of mass at the bottom of the inclined plane
(c) cylinder has the greatest velocity of the centre of mass at the bottom of the inclined plane
(d) all three reach the bottom with the same speed
Show Answer & Explanation

Answer: (a) ring has the least velocity of the centre of mass at the bottom of the inclined plane

Explanation:
1. v² = 2gh/(1 + k²/R²).
2. The ring has the largest k²/R² (1), so it has the least speed; the sphere (2/5) has the greatest.

Teacher's Note:
1. Smaller k²/R² means faster rolling.

Question 5: The centre of mass of a body is the average position of its
(a) mass
(b) weight
(c) either mass or weight
(d) none of these
Show Answer & Explanation

Answer: (a) mass

Explanation:
1. The centre of mass is the mass-weighted average position of all the particles.
2. It is defined through mass, not weight.

Teacher's Note:
1. CM is a mass-weighted average.

Question 6:
Assertion (A): A rigid disc rolls without slipping on a fixed rough horizontal surface with uniform angular velocity. Then the acceleration of the lowest point of the disc is zero.
Reason (R): For a rigid disc rolling without slipping on a fixed rough horizontal surface, the velocity of the lowest point on the disc is always zero.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (d) A is false, but R is true.

Explanation:
1. The lowest point is instantaneously at rest, but it has centripetal acceleration ω²R directed upward.
2. So the assertion is false and the reason is true.

Teacher's Note:
1. Zero velocity does not mean zero acceleration.

Question 7: A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and the other hanging freely. The tension in the string required to produce an angular acceleration of 2 s⁻² is
(a) 25 N
(b) 50 N
(c) 78.5 N
(d) 157 N
Show Answer & Explanation

Answer: (a) 25 N

Explanation:
1. I = ½MR² = ½ × 50 × 0.25 = 6.25 kg m².
2. τ = Iα = 12.5 N m = T × R, so T = 12.5/0.5 = 25 N.

Teacher's Note:
1. Torque = tension × radius.

Question 8: Two objects P and Q initially at rest move towards each other under mutual force of attraction. At the instant when the velocity of P is v and that of Q is 2v, the velocity of the centre of mass of the system is
(a) v
(b) 3v
(c) 2v
(d) zero
Show Answer & Explanation

Answer: (d) zero

Explanation:
1. The system is isolated and starts at rest, so the total momentum is zero.
2. The CM velocity remains zero.

Teacher's Note:
1. Internal forces keep the CM at rest.

Question 9: When sand is poured on a rotating disc, its angular velocity will
(a) decrease
(b) increase
(c) remain constant
(d) none of the above
Show Answer & Explanation

Answer: (a) decrease

Explanation:
1. Poured sand adds to the moment of inertia.
2. With no external torque, Iω is constant, so ω decreases.

Teacher's Note:
1. I increases, so ω decreases.

Question 10: The motion of the centre of mass depends on
(a) total external forces
(b) total internal forces
(c) the sum of (a) and (b)
(d) none of these
Show Answer & Explanation

Answer: (a) total external forces

Explanation:
1. M a⃗_cm = F⃗_ext.
2. Internal forces cancel in pairs and cannot affect the motion of the centre of mass.

Teacher's Note:
1. Only external forces move the CM.

Question 11: The angular momentum of a system of particles is conserved
(a) when no external force acts upon the system
(b) when no external torque acts upon the system
(c) when no external impulse acts upon the system
(d) when the axis of rotation remains the same
Show Answer & Explanation

Answer: (b) when no external torque acts upon the system

Explanation:
1. dL/dt = τ_ext.
2. L is conserved when the net external torque is zero.

Teacher's Note:
1. No torque, no change in L.

Question 12:
Assertion (A): The radius of gyration of a body is a constant quantity.
Reason (R): The radius of gyration of a body about an axis of rotation may be defined as the root mean square distance of the particles from the axis of rotation.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (d) A is false, but R is true.

Explanation:
1. The radius of gyration changes with the axis, so the assertion is false.
2. The reason gives the correct definition.

Teacher's Note:
1. k depends on the axis.

Question 13: Three masses are placed on the x-axis: 300 g at the origin, 500 g at x = 40 cm and 400 g at x = 70 cm. The distance of the centre of mass from the origin is
(a) 40 cm
(b) 45 cm
(c) 50 cm
(d) 30 cm
Show Answer & Explanation

Answer: (a) 40 cm

Explanation:
1. x_cm = (300 × 0 + 500 × 40 + 400 × 70)/1200.
2. = (20000 + 28000)/1200 = 40 cm.

Teacher's Note:
1. Take the mass-weighted mean of positions.

Question 14: The moment of inertia of a uniform circular disc of radius R and mass M about an axis passing through the edge of the disc and normal to the disc is
(a) \( MR^2 \)
(b) \( \frac12MR^2 \)
(c) \( \frac32MR^2 \)
(d) \( \frac72MR^2 \)
Show Answer & Explanation

Answer: (c) \( \frac32MR^2 \)

Explanation:
1. About the centre I = ½MR².
2. Parallel axis theorem: I = ½MR² + MR² = (3/2)MR².

Teacher's Note:
1. Add MR² for the shift to the edge.

Question 15:
Assertion (A): The centre of mass of a body may lie where there is no mass.
Reason (R): The centre of mass of a body is a point where the whole mass of the body is supposed to be concentrated.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. For a ring or a hollow sphere the CM lies in empty space.
2. This is because the CM is only a representative point at which the whole mass can be taken to act.

Teacher's Note:
1. CM is a representative point.

Question 16: A rigid body of mass M and radius R rolls without slipping on an inclined plane of inclination θ. Match the type of body in Column I with the magnitude of the force of friction in Column II.
Column IColumn II
(A) Ring(1) Mg sinθ/2.5
(B) Solid sphere(2) Mg sinθ/3
(C) Solid cylinder(3) Mg sinθ/3.5
(D) Hollow spherical shell(4) Mg sinθ/2

(a) A-4, B-3, C-1, D-2
(b) A-3, B-4, C-2, D-1
(c) A-4, B-2, C-3, D-1
(d) A-4, B-3, C-2, D-1
Show Answer & Explanation

Answer: (d) A-4, B-3, C-2, D-1

Explanation:
1. Friction f = Mg sinθ · (k²/R²)/(1 + k²/R²).
2. Ring: 1/2 (A-4). Solid sphere: 2/7 = 1/3.5 (B-3). Solid cylinder: 1/3 (C-2). Hollow sphere: 2/5 = 1/2.5 (D-1).

Teacher's Note:
1. Friction depends on k²/R².

Question 17:
Assertion (A): The position of the centre of mass of a body depends upon the shape and size of the body.
Reason (R): The centre of mass of a body always lies at the centre of the body.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (c) A is true, but R is false.

Explanation:
1. The CM depends on the distribution of mass, so the assertion is true.
2. It does not always lie at the geometric centre (for example in a cone), so the reason is false.

Teacher's Note:
1. CM is not always at the centre.

Question 18: A body A of mass M, while falling vertically downwards under gravity, breaks into two parts: a body B of mass M/3 and a body C of mass 2M/3. The centre of mass of bodies B and C taken together shifts compared to that of body A
(a) does not shift
(b) depends on the height of breaking
(c) towards body B
(d) towards body C
Show Answer & Explanation

Answer: (a) does not shift

Explanation:
1. The breaking is caused by internal forces, so the CM motion is unchanged.
2. The CM of B and C together continues as if A were intact.

Teacher's Note:
1. Internal forces do not shift the CM.

Question 19: The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is
(a) 5 : 7
(b) 2 : 3
(c) 2 : 5
(d) 7 : 5
Show Answer & Explanation

Answer: (a) 5 : 7

Explanation:
1. Rolling: a = g sin θ/(1 + 2/5) = (5/7)g sin θ.
2. Slipping (smooth): a = g sin θ.
3. Ratio = 5 : 7.

Teacher's Note:
1. Rolling reduces the acceleration.

Question 20: Point masses of 1, 2, 3 and 4 kg lie at the points (0, 0, 0), (2, 0, 0), (0, 3, 0) and (−2, −2, 0) respectively. The moment of inertia of this system about the X-axis will be
(a) 43 kg m²
(b) 34 kg m²
(c) 27 kg m²
(d) 72 kg m²
Show Answer & Explanation

Answer: (a) 43 kg m²

Explanation:
1. Distance from the x-axis equals |y|.
2. I = 1×0 + 2×0 + 3×3² + 4×2² = 27 + 16 = 43 kg m².

Teacher's Note:
1. I about the x-axis uses y² only.

Question 21: Which of the following is an essential condition for mechanical equilibrium of a body?
(a) Total force on the body should be zero
(b) Total torque on the body should be zero
(c) Both (a) and (b)
(d) Total linear momentum should be zero
Show Answer & Explanation

Answer: (c) Both (a) and (b)

Explanation:
1. Translational equilibrium needs ΣF = 0 and rotational equilibrium needs Στ = 0.
2. Both are required.

Teacher's Note:
1. Two conditions for equilibrium.

Question 22: An automobile moves on a road with a speed of 54 km/h. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m². If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is
(a) \( 8.58\ \text{kg m}^2\text{s}^{-2} \)
(b) \( 10.86\ \text{kg m}^2\text{s}^{-2} \)
(c) \( 2.86\ \text{kg m}^2\text{s}^{-2} \)
(d) \( 6.66\ \text{kg m}^2\text{s}^{-2} \)
Show Answer & Explanation

Answer: (d) \( 6.66\ \text{kg m}^2\text{s}^{-2} \)

Explanation:
1. v = 54 km/h = 15 m/s, so ω = 15/0.45 = 33.3 rad/s.
2. α = 33.3/15 = 2.22 rad/s².
3. τ = Iα = 3 × 2.22 = 6.66 kg m² s⁻².

Teacher's Note:
1. Convert units, then use τ = Iα.

Question 23:
Assertion (A): For a system of particles under a central force field, the total angular momentum is conserved.
Reason (R): The torque acting on such a system is zero.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. A central force passes through the fixed centre, so its torque about that centre is zero.
2. Hence the angular momentum is conserved.

Teacher's Note:
1. Central force means zero torque.

Question 24: A wheel is rotating at 1800 rpm about its own axis. When the power is switched off, it comes to rest in 2 minutes. Then the angular retardation in rad s⁻² is
(a) \( 2\pi \)
(b) \( \pi \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{\pi}{4} \)
Show Answer & Explanation

Answer: (c) \( \frac{\pi}{2} \)

Explanation:
1. ω₀ = 1800 × 2π/60 = 60π rad/s.
2. α = 60π/120 = π/2 rad/s².

Teacher's Note:
1. α = ω/t.

Question 25:
Assertion (A): When you lean behind over the hind legs of a chair, the chair falls back after a certain angle.
Reason (R): A centre of mass lying outside the system makes the system unstable.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. The chair tips when the vertical line through its CM leaves the base of support.
2. So the reason explains the assertion.

Teacher's Note:
1. Stability needs the CM above the base.

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