A mass–spring oscillator of amplitude moves on a frictionless surface

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

Problem

A mass–spring oscillator of amplitude

A

moves on a frictionless surface. The motion is described in one dimension along the

x

-axis with equilibrium at

x=0

. (a) Derive an expression for the kinetic energy

K

and the elastic potential energy

U

in terms of the displacement

x

, the amplitude

A

, and the total mechanical energy

E_{\text{total}}

. (b) Using your expressions, show algebraically that

K=U

when

|x|=\dfrac{A}{\sqrt{2}}

. (c) Explain why the result in part (b) is independent of the mass of the oscillator and the spring constant. (d) Sketch on separate axes the functional dependence of the fractions

K/E_{\text{total}}

and

U/E_{\text{total}}

on the dimensionless displacement ratio

x/A

for the interval

-1\le x/A \le 1

. Label key points found in part (b).

Related theory

SHM Energy
Energy in SHM
Energy in simple harmonic motion
Simple harmonic motion
Oscillations

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