Find the amplitude.?

Question

A frictionless block of mass 2.25 kg is attached to an ideal spring with force constant 290 N/m . At t=0 the spring is neither stretched nor compressed and the block is moving in the negative direction at a speed of 11.0 m/s.
a) Find the amplitude.
b) Find the phase angle. (In rad)

Answer 1

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Mass = m = 2.25 kg

Force constant = k = 290 N/m

Angular frequency = w = sq rt ( k/m) =sq rt (290/2.25 )=11.353 /s

Amplitude = A

At t=0 the spring is neither stretched nor compressed means block is at mean position and velocity is maximum and equal to 11.0 m/s

Maximum velocity =Vm = Aw =11

A =11/w = 11 /11.353=0.9689 m

(a) The amplitude is 0.9689 m
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As the block is moving in the negative direction while at mean position means half cycle is just completed and phase is 180 degree or ' pi ' radian

(b) Phase is pi radian
where pi = 22 / 7=3.1416
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Answer 2

Amplitude is easy enough. At the beginning of the problem all of the energy is kinetic:

E = 1/2 m v^2

Later, the block will stop for a split second and all of the mechanical energy is potential spring energy

E = 1/2 k A^2

Put them together and solve for amplitude A.

This occurs at π/2 radians or 90 degrees phase angle.