Conservation of Energy

Q. A flexible but inextensible chain of length $l$ and weight $w$ N/m is held on a smooth table with an initial overhang 'a' as shown in figure. What will be the velocity $v$ with which the chain will leave the table if released?

1. $v=\sqrt{\frac{g}{l}(l^2+a^2})$
2. $v=\sqrt{gl}$
3. $v=\sqrt{\frac{g}{l}\left(a^2\right)}$
4. $v=\sqrt{\frac{g}{l}(l^2-a^2})$

Q. A point mass is shot vertically up from ground level with a velocity of 4 m/s at time, t = 0. It loses 20% of its impact velocity after each collision with the ground. Assuming that the acceleration due to gravity is $10{\text{m/s}}^2$ and that air resistance is negligible, the mass stops bouncing and comes to complete rest on the ground after a total time (in seconds) of

1. 1
2. 2
3. 4
4. $\infty$

Q. A stone with mass of 0.1 kg is catapulted as shown in the figure. The total force $F_x$ (in N) exerted by the rubber band as a function of distance x (in m) is given by $F_x=300x^2$. If the stone is displaced by 0.1 m from the un-stretched position the rubber band is

1. 0.01 J
2. 0.1 J
3. 1 J
4. 10 J

Q. A point mass M is released from rest and slides down a spherical bowl of radius R from a height H as shown in the figure below. The surface of the bowl is smooth (no friction). The velocity of mass at the bottom of the bowl is

1. $\sqrt{gH}$
2. $\sqrt{2gR}$
3. $\sqrt{2gH}$
4. 0

Q. A block weighing 10 N travels down a smooth curved track AB joined to a rough horizontal surface as shown in figure below. The rough surface has a friction coefficient of 0.20 with the block. If the block starts slipping on the track from a point 1.0 m above the horizontal surface, how far will it move on the rough surface? (Assume g = 10 m/sec${}^2$)

1. 10 m
2. 12 m
3. 2 m
4. 5 m

Q. Two disc A and B with identical mass (m) and radius (R) are initially at rest. They roll down from the top of identical inclined planes without slipping disc A has all of its mass concentrated at the rim, while Disc B has its mass uniformly distributed. At the bottom of the plane, the ratio of velocity of the center of disc A to the velocity of the center of disc B is

1. $\sqrt{\frac{3}{4}}$
2. $\sqrt{\frac{3}{2}}$
3. 1
4. $\sqrt{2}$

Q. The figure below shows a small spherical ball of mass m rolling down the loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R. The tangential acceleration of the centre when the ball is at A is (neglect the radius of mass 'm')

1. $-\frac{5}{7}gcos\theta$
2. $-\frac{5}{7}gsin\theta$
3. $-\frac{5}{9}gcos\theta$
4. $-\frac{5}{9}gsin\theta$

Q. Consider the following two statements:

Statement (I): A body of weight h and strikes the ground. If the body starts from rest, the velocity with which it strikes the fround is $\sqrt{2gh}$, where 'g' is the acceleration due to gravity.



Statement (II): If the same boy (initially at rest) slides without friction along an inclined plane PQ (angle of inclination $\alpha$) starting from an elevation h above point Q, then its velocity at point Q is $\sqrt{2gh}$



The correct option is

1. Both statements 1 and 2 are true
2. Statements 1 is true and statement 2 is false
3. Statement 1 is fasle and statement 2 is true
4. Both statements 1 and 2 are false

Q. Two metallic blocks having masses in the ratio 2:3 are made to slide down a frictionless inclined plane starting initially from rest position . When these blocks reach the bottom of the inclined plane, they will have their kinetic energies in the ratio

1. 2:3
2. 3:5
3. 3:2
4. 7:4

Q. A stone tied to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position, and has a speed u. The magnitude of the change in its velocity as it reaches a position where the string is horizontal is

1. $\sqrt{u^2-2gL}$
2. $\sqrt{2gL}$
3. $\sqrt{u^2-gL}$
4. $\sqrt{2(u^2-gL)}$

Q. The 2 kg block shown in figure (top view) rests on a smooth horizontal surface and is attached to a massless elastic cord that has a stiffness $5\frac{N}{m}$

The cord hinged at O is initially unstretched and always remains elastic. The block is given a velocity v of $1.5\frac{m}{s}$ perpendicular to the cord. The magnitude of velocity in $\frac{m}{s}$ of the block at the instant the cord is stretched by 0.4 m is

1. 1.36
2. 1.07
3. 0.83
4. 1.50

Q. A circular cylinder of radius r and mass m, starts from the top of an inclined plane and rolls down without any slip. fter its center moves to a point having a vertical height h as shown below, the velocity of the center of mass, with g as acceleration due to gravity is

1. $\sqrt{\frac{gh}{3}}$
2. $\sqrt{\frac{2gh}{3}}$
3. $\sqrt{\frac{4gh}{3}}$
4. $\sqrt{2gh}$