# Numerical Problems , Laws of Motion

#### LEVEL – I

Q:1. Two blocks A and B of masses M1 and M2 respectively kept in contact with each other on a smooth horizontal surface. A constant horizontal force (F) is applied on ‘ A ‘ as shown in figure. Find the acceleration of each block and the contact force between the blocks

[Ans: a= F/( M1 + M2) , N=M2F/( M1 + M2) ]

Q:2. A bob of mass m = 50 gm is suspended from the ceiling of a trolley by a light inextensible string. If the trolley accelerates horizontally, the string makes an angle θ = 30° with the vertical. Find the acceleration of the trolley.

[Ans: 5.7 m/s2]

Q:3. Two small bodies connected by a light inextensible string passing over a smooth pulley are in equilibrium on a fixed smooth wedge as shown in the figure. Find the ratio of the masses. Given that θ = 60° and α = 30°

[Ans: m1/m2 = 1/√3]

Q:4. Both the springs shown in Figure are unstretched. If the block is displaced by a distance x and released, what will be the initial acceleration?

[Ans: 2kx/m]

Q:5. A block of mass m = 1 kg is at rest on a rough horizontal surface having coefficient of static friction 0.2 and kinetic force 0.15. Find the frictional forces if a horizontal force (a) F = 1 N, (b) F = 1.96 N and (c) F = 2.5 N are applied on a block which is at rest on the surface.

[Ans: (a) 1N (b) 1.96 N (c) 1.5 N ]

Q:6. Two masses m1 = 5 kg, m2 = 2 kg placed on a smooth horizontal surface are connected by a light inextensible string. A horizontal force F = 1 N is applied on m1. Find the acceleration of either block. Describe the motion of m1 and m2 if the string breaks but F continues to act.

Ans : [a = 1/7 m/s2 , a’ = 1/5 m/s2]

Q:7. The coefficient of static friction between a block of mass m and an incline is μs = 0.3.
(a) What can be the maximum angle θ of the incline with the horizontal so that the block does not slip on the plane?
(b) If the incline makes an angle θ/2 with the horizontal, find the frictional force on the block.

Ans: [θ = tan-1(0.3) , f = 0.145 mg ]

Q:8. A 20 kg box is dragged across a rough level floor having a coefficient of kinetic friction of 0.3 by a rope which is pulled upward at angle of 30° to the horizontal with a force of magnitude 80 N.

(a) What is the normal force?

(b) What is the frictional force?

(c) What is the acceleration of the box?

(d) If the force is reduced until the acceleration becomes zero, what is the tension in the rope?

Ans: [(a)160 N (b)48 N (c) 1.06 m/s2 (d) 55.42 N ]

Q:9. A small body A starts sliding down from the top of a wedge (fig.) whose base is equal to l = 2.10 m. The coefficient of friction between the body and the wedge surface is k = 0.140. For what value of the anlge α will the time of sliding be the least? What will it be equal to?

Ans : $\displaystyle \frac{2l}{((g/2)\sqrt{1 + k^2 – k)^{-1}}}$

Q:10. A chain of length l is placed on a smooth spherical surface of radius R with one of its ends fixed at the top of the sphere. What will be the acceleration a of each element of the chain when its upper end is released? It is assumed that the length of the chain l < (πR/2).

Ans: Tangential acceleration $\displaystyle a_t = \frac{R g}{l}(1- cos\frac{l}{R})$

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