# Numerical Problems , Work , Power , Energy

#### LEVEL – I

Q:1. An object of mass 5 kg falls from rest through a vertical distance of 20 m and attains a velocity of 10 m/s. How much work is done by the resistance of the air on the object? (g = 10m/s2).

Ans: -750 J

Q:2. A train of mass 100 metric tons is drawn up an incline of 1 in 49 at the rate of 36 km per hour by an engine. If the resistance due to friction be 10 N per metric ton, calculate the power of the engine. If the steam is shut off, how far will the train move before it comes to rest?

Ans: 238 m

Q:3. Shown in the figure is a smooth vertical frame of wire along which a small bead moves from the point A. Find its speed at the point B.

Ans : $\displaystyle v = \sqrt{2 g(H-h)}$

Q:4. A rubber ball after falling through a height h penetrates into the water through a distance x. Find the average force imparted by water on the rubber ball in ideal conditions.

Ans : $\displaystyle F = – mg (\frac{h}{x} + 1)$

Q:5. A bus of mass 1000 kg has an engine which produces a constant power of 50 kW. If the resistance to motion, assumed constant is 1000 N, find the maximum speed at which the bus can travel on level road and the acceleration when it’s travelling at 25 m/s.

Ans : v = 50 m/s ; a = 2 m/s2

Q:6. A particle of mass m is projected up the smooth inclined plane of inclination θ with a speed v0( > √(2glSinθ) ) as shown in the figure. Find the maximum height travelled by the particle.

Ans : $\displaystyle H = \frac{(v_0^2 – 2 g l sin\theta)sin^2 \theta}{2 g} + l sin\theta$

Q:7. A block of mass m collides with a horizontal weightless spring of force constant k. The block compresses the spring by x. Calculate the maximum momentum of the block.

Ans : $\displaystyle (\sqrt{k m} ) x$

Q:8. Prove that K. E. of two identical trains with respect to heliocentric frame of reference moving in opposite directions on equatorial line with same speed (w.r.t earth) are not equal.

Q:9. A small mass m starts from rest and slides down the smooth spherical surface of R. Assume zero potential energy at the top.
Find (a) the change in potential energy
(b) the kinetic energy
(c) the speed of the mass as a function of the angle made by the radius through the mass with the vertical.

Ans : (a) $\displaystyle -m g R(1-cos\theta)$
(b)$\displaystyle m g R(1-cos\theta)$
(c)$\displaystyle \sqrt{2 g R(1-cos\theta)}$

Q:10. An ideal massless spring can be compressed by 1 m by a force of 100 N. This same spring is placed at the bottom of a frictionless inclined plane which makes an angle θ = 30° with the horizontal. A 10 kg mass is released from rest at the top of the incline and is brought to rest momentarily after compressing the spring 2 meters.

(a) Through what distance does the mass slide before coming to rest?

(b) What is the speed of the mass just before it reaches the spring?

Ans: (a) s = 4 m (b) v = 2√5 m/s

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