26. Show that at any instant of time during the motion total mechanical energy of a freely falling body remains constant. Show graphically the variation of K.E. and P.E. during the motion.
27. Define spring constant, Write the characteristics of the force during the elongation of a spring. Derive the relation for the PE stored when it is elongated by X. Draw the graphs to show the variation of P.E. and force with elongation.
28. How does a perfectly inelastic collision differ from perfectly elastic collision? Two particles of mass m1 and m2 having velocities U1 and U2 respectively make a head on collision. Derive the relation for their final velocities. Discuss the following special cases.
(i) m1 = m2
(ii) m1 >> m2 and U2 = 0 (iii) m1 << m2 and u1 = 0
29. A body is moving along z-axis of a coordinate system under the effect of a constant force F
2i 3jk N
. Find the work done by the force in moving the body a distance of 2 m along z-axis.
30. In lifting a 10 kg weight to a height of 2m, 230 J energy is spent. Calculate the acceleration with which it was raised?
31. A bullet of mass 0.02 kg is moving with a speed of 10m–1s. It can penetrate
10 cm of a wooden block, and comes to rest. If the thickness of the target would be 6 cm only find the KE of the bullet when it comes out.
(Ans : 0.4 J) 32. A man pulls a lawn roller with a force of 20 kg F. If he applies the force
at an angle of 60° with the ground. Calculate the power developed if he takes 1 min in doing so.
33. A ball bounces to 80% of its original height. Calculate the mechanical energy lost in each bounce.
34. A pendulum bob of mass 0.1 kg is suspended by a string of 1 m long. The bob is displaced so that the string becomes horizontal and released. Find its kinetic energy when the string makes an angle of (i) 0°, (ii) 30° with the vertical.
35. A particle of mass m is moving in a horizontal circle of radius r under a centripetal force equal to k/r2, k is a constant. What is the total energy of
the particle. (–k/2r)
36. The K.E. of a particle moving along a circle of radius R depends on the distance covered S as T = S2 where is constant. Find the force acting
on the particle as a function of S. F 2 S 1 s2 R2 37. Derive the relation E = mc2.
Solution: As m mo 1 v2 c2 or m(1 – v2/c2)1/2 = m o or m 2(1 – v2/ c2) = m o 2 or m2c2 – m2v2 = m o 2c2
Differentiating both sides taking mo and c as constant
But 2m 0 so, dm c2 –dm v2 – m v dv = 0 or v2dm + m v dv = c2 dm (1)
Let a force F applied on the body displaces it through a distance dx. The work done in doing so will change the KE of the body. K.E.=dE =F.dx But F = d(mv)/dt = (mdv/dt + vdm/dt) or dw = F.dx dw = (mdv/dt + vdm/dt)dx or dw = mdv/dt. dx + vdm/dt dx dw = mdx/dt.dv + vdx/dt dm dw = dE = m v dv + v2 dm
from equ. (i) dE = c2 dm (2)
let the mass increases from mo to m as K.E. increases from 0 to E. Integrating eq. 2 for the given limits
E – o = c2(m – m
o) As mo c2 rest mass energy = o
therefore E = m c2
38. Force acting on a particle varies with x as shown in figure, below. Calculate the work done by the force as the particle moves from x = 0 to x = 6m.
39. The mass system is kept on sphere. Ball 1 is slightly disturbed. What is the velocity of these balls when it is making angle ‘’ with horizontal (friction is absent everywhere).
40. What is the minimum value of ‘u’ for completing circular motion of particle as shown in figure given below?
u
41. A block is projected horizontally on rough horizontal floor. The coefficient of friction between the block and the floor is . The block strikes a light spring of stiffness k with velocity v0. Find the maximum compression of the spring.
42. In figure a and b, AC, DE and EF are fixed inclined planes BC = EF = x and AB = DE = y. A small block of mass m is released from rest from the point A. It slides down AC and reaches C with a speed VC. The same block is released from rest from the point D, it slides down DEF and reaches the point F with speed VF. The coefficient of kinetic friction between the block and the surface AC and DEF is a, calculate VC and VF.
43. A locomotive of mass m starts moving so that its velocity v is according to the law va s, where a is constant and s is distance covered. Find the total work done by all the forces acting the locomotive during the first t seconds after the beginning of motion.
44. The Kinetic energy of a particle moving along a circle of radius R depends on the distance covered S and T = s2 where is constant. Find the force
acting on the particle as a function of S.
45. A ball suspended by a string of length 20 cm is fixed to the free end of the pivoted rod of length 40 cm as shown in the figure. The rod is made to rotate in a horizontal plane with constant angular speed. The string makes an angle = 30° with the vertical axis. Find the angular speed of the rotation?
= angular speed; T = Tension in the string.
46. A smooth, light rod AB can rotate about a vertical axis passing through its end A. The rod is fitted with small sleeve of mass m attached to the end A by a weightless spring of length 10, stiffness k. What work must be performed to slowly get this system going and the angular velocity w?
47. A spring gun having spring constant 100 N/m, a small ball of mass 0.1 Kg is placed in its barrel by compressing the spring through 0.05m as shown in figure.
(a) Find the velocity of the ball when spring is released.
(b) Where should a box be placed on ground so that ball falls in it, if the ball leaves the gun horizontally at a height of 2m above the ground.
48. A turn of radius 20m is banked for the vehicle of mass 200kg going at a speed of 10ms–1. Find the direction and magnitude of frictional force acting
on a vehicle if it moves with a speed (a) 5 m/s (b) 15 m/s assume the friction is sufficient to prevent slipping (g = 10m/s2).
49. A block of mass m is pushed against a spring constant K fixed at one end to a wall. The block can slide on a frictionless table as shown in figure. The natural length of the spring is L0 and it is compressed to half its natural length when the block is released. Find the velocity of the block as a function of its distance x from the wall.
L0/2 x
50. A ball falls under gravity from a height 10m, with an initial velocity V0, it hits the ground, looses 50% of its energy after collision and it rises to the same height. What is the value of V0?
51. A block of mass M is pulled along a horizontal surface by applying a force at an angle with horizontal. Co-efficient of friction between block and surface is . If the block travels with uniform velocity, Find the work done by this applied force during a displacement d of the block.
52. A block of mass m released from rest onto an ideal non-deformed spring of spring constant ‘K’ form a negligible height. Neglecting the air resistance, find the compression ‘d’ of the spring.
53. A particle slides down a smooth inclined plane of elevation , fixed in an elevator going up with an acceleration a0 as shown in figure. The base of the incline has a length L. Find the time taken by the particle to reach the bottom.
54. 6m long ladder weighting 30Kg rest with its upper end against a smooth wall and lower end on rough ground. What should be the minimum coefficient of friction between the ground and the ladder for it to be inclined at 60° with the horizontal without slipping? Take g = 10m/s2.
55. A block of mass 200kg is set into motion on a frictionless horizontal surface with the help of frictionless pulley and a rope system as shown in figure. What horizontal force F should be applied to produce in the block an acceleration of 1 m/s2?
56. Two bodies of masses m1 and m2 are connected by a light string going over a smooth light pulley at the end of an incline. The mass m1 lies on the incline and m2 hangs vertically. The system is at rest; Find the angle of incline and the force exerted by the incline on the body of mass m1.