Amplitude Change in Oscillations with Varying Spring Constants

[morrisj753]

Homework Statement

A mass is attached to the wall by a spring of constant k. When the spring is at its natural length, the mass is given a certain initial velocity, resulting in oscillations of amplitude A. If the spring is replaced by a spring of constant 2k, and the mass is given the same initial velocity, what is the amplitude of the resulting oscillation?

Homework Equations

E = 1/2mv^2 + 1/2kx^2 = 1/2kA^2

The Attempt at a Solution

I am not quite sure what equation to really apply here, or how to approach the problem to find the amplitude. The answer is 1/(√2).

 

[ehild]

The velocity is given when the spring is relaxed. So you get the amplitude from the equation 1/2mv^2 =1/2kA^2

Double the spring constant: k'=2k. Now you have the equation for the new amplitude A'. 1/2mv^2 =1/2 k' A'^2 . V is the same, so 1/2kA^2= 1/2 k' A'^2. How are A and A' related?


ehild

 

[morrisj753]

well, by substituting k' = 2k,
you end up with A^2 = 2 (A')^2

 

[ehild]

Isolate A'.

ehild

 

[morrisj753]

You end up with 1/√2 A, which is the correct answer.
Thank you very much for the help!