Match the Following
Q.1. A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U0 constants). Match the potential energies in column I to the corresponding statement(s) in column II.
|Column I||Column II|
|(p) The force acting on the particle is zero at x = a|
|(q) The force acting on the particle is zero at x = 0|
r) The force acting on the particle is zero at x = – a
(s) The particle experiences an attractive force towards x = 0 in the region |x| < a
(t) The particle with total energy U0/4
can oscillate about the point x = – a
Ans. (A) p, q, r, t; (B) q, s; (C) p, q, r, s; (D) p, r, t
Solution. A → p, q, r, t; B → q, s; C → p, q, r, s; D → p, r, t
Integer Value Correct Type:-
Q.1. A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. (2009)
Given m = 0.36 kg, M = 0.72 kg.
The figure shows the forces on m and M. When the system is released, let the acceleration be a. Then T – mg = ma Mg – T = Ma
and T = 4 mg/3
For block m :
u = 0, a = g/3, t = 1, s = ?
∴ Work done by the string on m is
Q.2. Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. There after, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C. (2009)
Ans. 4 m/s
Solution. The velocity of B just after collision with A is
The collision between B and C is completely inelastic.
Q.3. A block of mass 0.18 kg is attached to a spring of forceconstant 2 N/m. The coefficient of friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un-stretched. An impulse is given to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time. The initial velocity of the block in m/s is V = N/10. Then N is
Solution. Let v be the speed of the block just after impulse. At B, the block comes to rest. Therefore
Loss in K.E. of the block = Gain in P.E. of the spring + Work done against friction
Q.4. A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in ms–1) of the particle is zero, the speed (in ms–1) after 5 s is
Q.5. Consider an elliptical shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictionless losses, the kinetic energy of the block when it reaches Q is (n × 10) joules. The value of n is (take acceleration due to gravity = 10 ms–2)
Solution. Work done = Increase in potential energy + gain in kinetic energy
F × d = mgh + gain in K.E.
18 × 5 = 1 × 10 × 4 + gain in K.E.
∴ Gain in K.E. = 50 J = 10n
∴ n = 5