This question will look at the energy in an undamped oscillator

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

This question will look at the energy in an undamped oscillator. Consider a mass,

m

, oscillating vertically on a spring of spring constant

k

, with an amplitude of

x_o

. We have seen already that the velocity of the mass at any displacement is given by the equation

v = u \sqrt{x_o^2 - x^2} .

(1) Using the equation for

v

, given earlier, write an equation for the kinetic energy of the mass as a function of its displacement. Hence, what is the maximum kinetic energy of the mass? Where does this occur? (2) Now consider the potential energy on the springs. For an extension,

x

, write an equation for the potential energy stored in the stretched spring in terms of

k

and

x

. Hence, what is the maximum potential energy stored in the stretched spring? For the mass-spring system, you should have seen already that

\omega^2 = \frac{k}{m}

. Use this identity to show that the maximum potential energy stored in the stretched spring is equal to the maximum kinetic energy of the mass. (3) When the kinetic energy of the mass is a maximum, where is the mass? what is the value of the potential energy stored in the spring? (4) When the potential energy stored in the spring is a maximum, where is the mass? what is the value of the kinetic energy of the mass? (5) So, what can you say about the total energy of the mass-spring system?

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