Title 4-dynamics-of-a-particle.pdf ✅

Dynamics of a Particle 1. A block of mass m is placed on a surface with a vertical cross-section given by y = x 3 /6. If the coefficient of friction is 0.5 , the maximum height above the ground at which the block can be placed without slipping is : (A) 1 6 m (B) 2 3 m (C) 1 3 m (D) 1 2 m 2. Two billiard balls A and B, each of mass 50 g and moving in opposite directions with speed of 5 m s −1 each, collide and rebound with the same speed. If the collision lasts for 10−3 which of the following statements are true? (A) The impulse imparted to each ball is 0.25 kg ms−1 and the force on each ball is 250 N (B) The impulse imparted to each ball is 0.25 kg ms−1 and the force exerted on each ball is 25 × 10−5 N (C) The impulse imparted to each ball is 0.5Ns (D) The impulse and the force on each ball are equal in magnitude and opposite in direction 3. Two blocks A and B of the same mass are joined by a light string and placed on a horizontal surface. An external horizontal force P acts on A. The tension in the string is T. The forces of static friction acting on A and B are F1 and F2 respectively. The limiting value of F1 and F2 is F0. As P is gradually increased. (A) For P < F0, T = 0 (B) For F0 < P < 2F0, T = P − F0 (C) For P > 2F0, T = P/2 (D) None of the above 4. A rough vertical board has an acceleration ' a ' along the horizontal so that a block of mass m pressing against it, does not fall. The coefficient of friction between the block and board should be at least (A) = g/a (B) > g/a (C) < g/a (D) > a/g 5. A simple pendulum with a bob of mass m is suspended from the roof of a car moving with a horizontal acceleration a. The bob is at rest with respect to the car, then : (A) The string makes an angle of tan−1 (a/g) with the vertical (B) The string makes an angle of tan−1 (1 − a g ) with the vertical (C) The tension in the string is m√a 2 + g2 (D) The tension in the string is m√g2 − a 2 6. Two block of mass 4 kg and 5 kg attached a light spring are suspended by a string as shown in figure. Find acceleration of block 5 kg and 4 kg. Just after the string is cut : (A) 8 m/s 2 , 10 m/s 2 (B) 10 m/s 2 , 10 m/s 2 (C) 18 m/s 2 , 0 (D) 8m/s 2 , 0
7. A block of weight W is suspended from a spring balance. The lower surface of the block rests on a weighing machine. The spring balance reads W1 and the weighing machine reads W2. ( W, W1,W2 are in the same unit.) (A) W = W1 + W2 if the system is at rest (B) W > W1 + W2 if the system moves down with some acceleration (C) W1 > W2 if the system moves up with some acceleration (D) No relation between W1 and W2 can be obtained with the given description of the system 8. The motion of a particle of mass m is given by x = 0 for t < 0s, x(t) = Asin 4pt for 0 < t < (1/4)s(A > 0) and x = 0 for t > (1/4)s : (A) The force at t = (1/8)s on the particle is −16π 2Am (B) The particle is acted upon by on impulse of magnitude 4π 2 A m at t = 0 s and t = (1/4)s (C) The particle is not acted upon by any force (D) The particle is not acted upon by a constant force (E) There is no impulse acting on the particle 9. In fig. the co-efficient of friction between the floor and the body B is 0.1 . The co-efficient of friction between the bodies B and A is 0.2 . A force F is applied as shown on B. the mass of A is m/2 and of B is m. Which of the following statements are true? (A) The bodies will move together if F = 0.25mg. (B) The body A will slip with respect to B if F = 0.5mg. (C) The bodies will move together if F = 0.5mg. (D) The bodies will be at rest if F = 0.1mg. (E) The maximum value of F for which the two bodies will move together is 0.45mg. 10. Friction co-efficient between block and inclined surface is μs . This maximum value of M for which system will remain in equilibrium. (A) m(sin θ + μscos θ) (B) 2msin θ (C) 2m(sin θ + μscos θ) (D) msin θ 11. Figure shows a heavy block kept on a frictionless surface and being pulled by two ropes of equal mass m. At t = 0, the force on the left rope is withdrawn but the force on the right end continues to act. Let F1 and F2 be the magnitudes of the forces by the right rope and the left rope on the block respectively. (A) F1 = F2 = F for t < 0 (B) F1 = F2 = F + mg for t < 0 (C) F1 = F, F2 = F for t > 0 (D) F1 < F, F2 = F for t > 0 12. A particle O, of mass m is attached to a vertical rod with two inextensible strings AO and BO of equal lengths l. The distance between the points of suspension on the vertical rod is also l. If the setup rotates with angular frequency ω, then : (A) Tension in thread BO is greater (B) Tension in thread AO is greater (C) Tension in the two threads are equal (D) Tension in AO or BO is greater according as ω is anticlockwise or clockwise
13. A rectangular block weighing 150 N, is lying on a rough inclined plane with inclination angle 45∘ as shown in the figure. The block is tied up by a horizontal string which has a tension of 50 N to keep the block just in equilibrium, then the coefficient of friction between the block and the inclined surface is : (A) Zero (B) 0.33 (C) 0.5 (D) 0.7 14. Blocks A and C start from rest and move to the right with acceleration aA = 12t m/s 2 and aC = 3 m/s 2 . Here t is in seconds. The time when block B again comes to rest is : (A) 2s (B) 1s (C) 3/2s (D) 1/2s 15. Two blocks of masses m1 and m2 are connected with a massless spring and placed over a plank moving with an acceleration ' a ' as shown in figure. The coefficient of friction between the blocks and platform is μ. (A) Spring will be stretched if a > μg (B) Spring will be compressed if a < μg (C) Spring will neither be compressed nor be stretched for a < μg (D) Spring will be in its natural length under all conditions 16. A block of mass m is placed on a rough horizontal surface. The coefficient of friction between them is μ. An external horizontal force is applied to the block and its magnitude is gradually increased. The force exerted by the block on the surface is R. (A) The magnitude of R will gradually increase (B) R ≤ mg√μ 2 + 1 (C) The angle made by R with the vertical will gradually increase (D) The angle made by R with the vertical ≤ tan−1 μ 17. Two bodies of mass m and 4m are attached by a string shown in the figure. The body of mass m hanging from a string of length l is executing simple harmonic motion with amplitude A while other body is at rest on the surface. The minimum coefficient of friction between the mass 4 m and the horizontal surface must be : (A) 1 4 (1 − A 2 l 2 ) (B) 1 4 (1 + A 2 l 2 ) (C) 1 4 A l cos θ (D) 1 4 18. In the system shown in figure, m1, > m2 system is held at rest by thread BC. Just after the thread BC is burnt: (A) Acceleration of m2 will be upwards
(B) Magnitude of acceleration of both blocks will be equal to ( m1−m2 m1+m2 )g (C) Acceleration of mx will be equal to zero (D) Magnitudes of acceleration of two blocks will be non-zero and unequal 19. Pushing force that makes an angle θ with the horizontal is applied on a block of weight W placed on a horizontal table. If the angle of friction be λ, the magnitude of force required to move the body is equal to (A) Wcos λ cos (θ−λ) (B) Wsin λ cos (θ−λ) (C) Wtan λ cos (θ−λ) (D) Wsin λ gsin (θ−λ) 20. A block of mass 10 kg is placed in a box as shown in figure. Box is moving with constant acceleration of 5 m/s 2 at an angle of 53∘ from x axis (horizontal direction). Force exerted by box on block in y-direction (vertical direction) will be : (g = 10 m/s 2 , tan 53∘ = 4/3) (A) 140N (B) 40N (C) 50N (D) 150N 21. A particle of mass m is released from rest at point A along the inside surface of a smooth hemispherical bowl of radius R. The speed at B which is at a height h = 2R 3 from the lowest point is : (A) √2gR (B) √ 4gR 3 (C) √gR (D) √ 2 3 gR 22. For the system shown in the figure, the pulleys are light and frictionless. The tension in the string will be : (A) 2 3 mgsin θ (B) 3 2 mgsin θ (C) 1 2 mgsin θ (D) 2mgsin θ 23. The two blocks A and B of equal mass are initially in contact. When released from rest on the inclined plane, they slide down the incline.