Q60. On a level circular cycling track of radius 10 m, riders P, Q and R travel at 2, 3 and 5 m/s respectively. If tyre-road friction is 0.1 and g = 10 m/s², which rider slips?
[UPSC NDA-II, 13 Sep 2026]
(A) P only
(B) Q only
(C) R only
(D) All three riders
Answer: (C) R only
Explanation: Friction provides the centripetal force on a level road. The maximum safe speed is √(μrg) = √10, approximately 3.16 m/s. P and Q remain below this limit, while R requires greater friction than the road can supply and slips.
Q59. A tennis ball of mass 350 g is projected vertically upward at 4 m/s. What are its maximum gravitational potential energy M and maximum height H? Take g = 10 m/s².
[UPSC NDA-II, 13 Sep 2026]
(A) M = 5.6 J; H = 1.6 m
(B) M = 2.8 J; H = 1.6 m
(C) M = 5.6 J; H = 0.8 m
(D) M = 2.8 J; H = 0.8 m
Answer: (D) M = 2.8 J; H = 0.8 m
Explanation: At the top, the initial kinetic energy becomes gravitational potential energy. Thus M = ½ × 0.35 × 4² = 2.8 joules. Equating mgh to 2.8 gives H = 4²/(2 × 10) = 0.8 metre, neglecting air resistance.
Q58. Consider three statements: I. Weight is constant everywhere on Earth while inertial mass may vary. II. Inertial mass and gravitational mass are proportional with the same constant everywhere. III. Inertial mass and weight are the same quantity. Which is correct?
[UPSC NDA-II, 13 Sep 2026]
(A) I only
(B) II only
(C) I and II
(D) II and III
Answer: (B) II only
Explanation: Mass is an intrinsic property, while weight equals mg and changes with local gravitational acceleration. Inertial and gravitational masses are proportional, as established by the equivalence principle. Therefore I and III are false; only statement II gives the correct physical relationship.
Q57. Rain falls vertically with speed v. One observer travels east at speed v and another north at speed v. What angle separates the apparent directions of the rain observed by them?
[UPSC NDA-II, 13 Sep 2026]
(A) 0°
(B) 30°
(C) 45°
(D) 60°
Answer: (D) 60°
Explanation: Each observer subtracts their own horizontal velocity from the vertical rain velocity. The resulting relative velocity vectors have equal magnitudes √2v, while their dot product equals v². Therefore the cosine of the angle is one-half, giving an angle of 60°.
Q56. What minimum horizontal speed must a pendulum bob have at its lowest point to complete a vertical circle when its string is 50 cm long and g = 10 m/s²?
[UPSC NDA-II, 13 Sep 2026]
(A) 5 m/s
(B) 7.5 m/s
(C) 10 m/s
(D) 12.5 m/s
Answer: (A) 5 m/s
Explanation: At the highest point, minimum tension is zero, requiring top speed √(gL). From the bottom to the top, the bob rises by 2L. Energy conservation gives bottom speed √(5gL), equal to √25 or 5 m/s.
Q55. In carrom, a 15 g striker hits a stationary 5 g coin. The coin acquires 0.36 m/s in 3 milliseconds. What is the average force on the coin?
[UPSC NDA-II, 13 Sep 2026]
(A) 0.72 N
(B) 0.60 N
(C) 0.54 N
(D) 0.30 N
Answer: (B) 0.60 N
Explanation: Average force equals the coin’s change in momentum divided by contact time. Its momentum gain is 0.005 × 0.36 = 0.0018 kg m/s. Dividing by 0.003 second yields 0.60 N. The striker’s own mass is unnecessary.
Q54. Two astronauts float at rest in deep space. Astronaut X throws a tool towards astronaut Y. For the system consisting of both astronauts and the tool, what happens to its centre of mass?
[UPSC NDA-II, 13 Sep 2026]
(A) It moves towards Y
(B) It follows astronaut X
(C) It follows the tool
(D) It remains fixed
Answer: (D) It remains fixed
Explanation: No external force acts on the combined system. The tool gains momentum towards Y, while astronaut X recoils with equal opposite momentum. The total momentum, initially zero, remains zero. Hence the velocity and position of the system’s centre of mass remain unchanged.
Q53. A uniform rope hangs over a smooth pulley, and a small extra mass makes one side heavier. Which statement is correct after the rope begins moving?
[UPSC NDA-II, 13 Sep 2026]
(A) Every rope element has the same acceleration vector
(B) All elements have the same speed but not the same acceleration vector
(C) Tension is identical throughout the rope
(D) The rope’s centre of mass stays stationary
Answer: (B) All elements have the same speed but not the same acceleration vector
Explanation: Because the rope is inextensible, its elements have equal speed magnitudes. Their acceleration directions differ on opposite hanging sides, and the curved portion has centripetal acceleration. Since the rope has mass, tension may vary; the system’s centre of mass also moves.
Q52. A car increases its velocity from 10 m/s to 30 m/s in 5 seconds. What is its acceleration?
[RRB Group D, 5 Aug 2026]
(A) 2 m/s²
(B) 10 m/s²
(C) 6 m/s²
(D) 4 m/s²
Answer: (D) 4 m/s²
Explanation: Acceleration is the rate at which velocity changes. Here the change is 30 − 10 = 20 m/s, spread uniformly over five seconds. Dividing 20 by 5 gives 4 m/s². The positive sign indicates that the car is speeding up.
Q51. Find the linear momentum of a body of mass 10 kg moving at 50 m/s.
[RRB NTPC Undergraduate, 18 Jun 2026]
(A) 40 kg m/s
(B) 60 kg m/s
(C) 5 kg m/s
(D) 500 kg m/s
Answer: (D) 500 kg m/s
Explanation: Linear momentum is the product of mass and velocity. Using p = mv, a ten-kilogram body moving at fifty metres per second has momentum of 10 × 50 = 500 kg m/s. Momentum is directional, so it points along the body’s velocity.