A mass is attached to the end of a spring and set into simple harmonic motion wi

Question

A mass is attached to the end of a spring and set into simple harmonic motion with an amplitude A on a horizontal frictionless surface. Determine the following in terms of only the variable A.

(a) Magnitude of the position (in terms of A) of the oscillating mass when its speed is 50% of its maximum value.
_________A

(b) Magnitude of the position (in terms of A) of the oscillating mass when the elastic potential energy of the spring is 50% of the total energy of the oscillating system.
_________A

Explanation / Answer

a) position is given by x = Acos(wt)

then v= dx/dt = -wA cos(wt)

v = Vmax (coswt)

so at some position , v and x related as

x^2 = 1 - v^2

so when v = 0.50Vmaax

then x = A ( sqrt(1 - 0.50^2)) = 0.866 A

Ans. 0.866A

b) PE + KE = constant

when PE = 0.50PEmax

then KE = 0.50KEmax

kx^2 /2 = 0.50kA^2 /2

x = sqrt(0.50)A = 0.707A

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