# Numerical Problems :Work ,Power & Energy

#### LEVEL – II

Q:1. A spring of mass m and stiffness k is fitted to a block of mass M. The system is moving with a constant velocity v on a smooth horizontal surface. If the system collides with a wall, find the maximum compression of the spring before it recoils, assuming that the total energy is conserved. Ans : $\displaystyle x = \sqrt{\frac{(3M + m)}{3 k}} v$

Q:2. A body of mass m is pushed with the initial velocity v0 up an inclined plane of angle of inclination θ. The co- efficient of friction between the body and the plane is μ. What is the net work done by friction during the ascent of the body, comes to stop.

Ans: $\displaystyle W_f = -\frac{\mu m v_0^2}{2(tan\theta + \mu )}$

Q:3. A block of mass m slides from the top of an inclined plane of angle of inclination θ & length l. The coefficient of friction between the plane and the block is μ. Then it is observed cover a distance d along the horizontal surface having the same coefficient of friction μ, before it comes to a stop. Find the value of d.

Ans : $\displaystyle d = l(sin\theta – \mu cos\theta)/\mu$

Q:4. A particle of mass m moves along a straight line on smooth horizontal plane, acted upon by a force delivering a constant power P. If the initial velocity of the particle is zero, then find its displacement as a function of time t.

Ans: $\displaystyle (\frac{8P}{9m} t^3 )^{1/2}$

Q:5. The figure shows a ball A of mass m connected to a light spring of stiffness k. Another identical ball B is connected with the ball A by a light inextensible string as shown in the figure. Other end of the spring is fixed. Initially the spring is in relaxed position. A vertical force F acts on B such that the balls move slowly. What is the work done by the force in pulling the ball B till the ball A reaches at the top of the cylindrical surface. The ball A remains in contact with the surface and co-efficient of friction between the surface and the ball A is μ .

Ans : $\displaystyle mgR [\frac{\mu}{\sqrt{2}} -\frac{\pi}{4} +1 -\frac{1}{\sqrt{2}}]$

Q:6. A body of mass m was slowly hauled up the hill. (Fig.) by a force F which at each point was directed along a tangent to the trajectory. Find the work performed by this force, if the height of the hill is h, the length of its base l, and coefficient of kinetic friction k. Ans: mg(k l + h)

Q:7. A particle of mass m moves along a circle of radius R with a normal acceleration varying with time as wn = at2, where a is a constant. Find the time dependence of the power developed by all the forces acting on the particle, and the mean value of this power averaged over the first t seconds after the beginning of motion.

Ans : [(1/2)m a R t ]

Q:8. A bob hangs from a rigid support by an inextensible string of length l . If it is displaced through a distance l keeping the string straight & released, find the speed of the bob at the lowest position.

Ans: $\displaystyle v = \sqrt{g l}$

Q:9. A body is projected from the top of a smooth fixed frictionless semi circular vertical tube of radius R with a speed √gR . Find the speed of the body when it descends through a vertical distance R/2.

Ans: $\displaystyle v = \sqrt{2 g R}$

Q:10. Two blocks of masses m1, and m2 connected by a light spring of stiffness k, are kept on a smooth horizontal surface as shown in the figure. What should be the initial compression of the spring so that the system will be about to break off the surface, after releasing the block m1? Ans: $\displaystyle \frac{(2m_1 + m_2)g}{k}$

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