# Astronomy Olympiad

## Problems on Energy conservation

The law of conservation of energy tells us the total energy of the system is always constant unless added into it by external parameters. More precisely, in an isolated system the energy can be transformed from one form into another but the total energy of the system always remains constant.

Some important forms of energy are:-

1. Mechanical: - the sum of potential and kinetic energy of the system.

2.

Gravitational potential energy: - The energy change in the particle when it moves from one point to another under the influence of the gravity.

F=mg (h_{1}-h_{2}) OR F=mgh when the reference point is defined.

3. Elastic potential energy:- The work done in moving the particle from it’s equilibrium position to a point x while performing SHM.

Q1) A particle falls from a height x=10m. from rest under the influence of gravity. Calculate the gain in the kinetic energy of the particle.

Q2) As show in the figure a small frictionless particle of mass m slides along the frictionless loop the loop. The particle is released from rest at point P, at height h=5R above the bottom of the loop.

A1) How much the work the weight of the block do on the block as th block travels from point P to

i) A point Q and

ii) The top of the loop?

A2) If the gravitational potential energy of the block – earth system is taken to be zero at the bottom of the loop, what is the potential energy when the block is

i) At point P

ii) At point Q

iii) At the top of the loop.

Q3) As shown in the figure on left a circus beagle of mass 6kg runs onto the left end of a curved ramp with speed v0=7.8 m/s at height y0=8.5 m above the floor. It then slides to the right comes to a momentary stop when it reaches at a height of y=11.1 m from the floor. What is the increase in the thermal energy of the beagle and ramp due to sliding?

Q4) A spring with a spring constant of 3200N/m is initially stretched until the elastic potential energy is 1.44 J. (U=0 for no stretch) What is the change in the elastic potential energy if

A1) the initial stretch is changed to a stretch is changed to 2cm.

A2) Compression of 2cm.

A3) Compression of 4 cm.

(Hint: A compression will have a sign opposite to stretch.)