Two separate oscillators, X and Y

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

Problem

Two separate oscillators, X and Y, have the same total mechanical energy

E

, but differ in mass and amplitude. The data are: • Oscillator X: mass

m_X = 0.60\,\text{kg}

, amplitude

A_X = 0.03\,\text{m}

• Oscillator Y: mass

m_Y = 0.20\,\text{kg}

, amplitude

A_Y = 0.06\,\text{m}

Answer the following: (i) Show that the ratio of their angular frequencies is

\dfrac{\omega_X}{\omega_Y} = \dfrac{A_Y}{A_X}\sqrt{\dfrac{m_Y}{m_X}}

. (ii) Using the given values, calculate

\omega_X/\omega_Y

to two decimal places. (iii) Hence determine the ratio of their spring constants

k_X/k_Y

. (iv) Which oscillator experiences the larger maximum acceleration? Support your answer with appropriate reasoning and calculation.