A cart attached to a spring vibrates with amplitude A Part A

A cart attached to a spring vibrates with amplitude A.

Part A

What fraction of the total energy of the cart-spring system is kinetic energy when the cart is at position x=2A/5.

Express your answer using three significant figures.

Part B

At what position is the cart when its kinetic energy equals its elastic potential energy?

Express your answer in terms of A.

Solution

amplitude is A (maximum displacement from the equilibrium position)

therefore the total energy = ½kA² (velocity = 0 at this position)

k = spring constant

k.e = ½mv² = t.e - spring potential energy = ½kA² - ½k(2A/5)²

= ½kA²[1 - 4/25] = ½kA²[21/25]

fraction of the total energy of the cart-spring system is kinetic energy = ½kA²[21/25] / ½kA² = 21/25

ii) k.e = s.p.e

=> ½mv² + ½kx² = ½kA² = 2[½kx²]

or x = A/[2]