Damped Spring Oscillator Energy Analysis · BPhO 2007 #9
Physics · mechanics
Problem
A mass M is attached to the end of a horizontal spring. The mass is pulled to the right, 8 cm from its rest position. It is then released so that the mass oscillates to the left and right, with the system gradually losing energy over many cycles.
[Figure: Spring-mass system showing 8cm displacement in both directions]
a) State the energy changes that take place over one complete cycle as the mass moves to the left and then back to the right. [2]
b) The energy stored in a stretched spring is proportional to the square of the extension of the spring. If after some time, the amplitude of the oscillation is reduced to 1 cm, what fraction of the initial energy has been lost? Show your working. [2]
c) We will need to use a concept that you have met in radioactivity. State what is meant by the half-life of a radioactive substance. [1]
d) Now we shall apply this concept to the loss of energy from the oscillating system. The amplitude decays away in the same manner as radioactive decay (exponentially). How many half-lives have passed for the amplitude to reduce to 1 cm? [2]
e) The period of oscillation does not depend upon the amplitude of the oscillation, being the same for both large and small amplitudes. The period of oscillation is 0.5 seconds. The half-life for the amplitude loss is 5 seconds. How many oscillations have occurred by the time the amplitude has dropped down to 1cm? [2]
f) The energy is also dissipated away exponentially with time. Using your answer to part (b) for the energy lost, how many energy loss half-lives have passed when the amplitude has reduced to 1 cm? [2]