An object with mass 0.200 kg is acted on by an elastic restoring force with forc

Question

An object with mass 0.200 kg is acted on by an elastic restoring force with force constant 10.5 N/m . The object is set into oscillation with an initial potential energy of 0.130 J and an initial kinetic energy of 6.40×102 J .

What is the amplitude of oscillation?

What is the potential energy when the displacement is one-half the amplitude?

At what displacement are the kinetic and potential energies equal?

What is the value of the phase angle if the initial velocity is positive and the initial displacement is negative?

Explanation / Answer

When PEf = KEf

1/2kA^2 = PEf + KEf

1/2kA^2 = PEf + PEf

1/2kA^2 = 2PEf

1/2kA^2 = 2*1/2kx^2

1/2kA^2 = kx^2

1/2A^2 = x^2

x = A/sqrt2 = 0.192/sqrt2 = 0.134m = 13.4cm

initial velocity , vi

KEi = 1/2mvi^2

6.40*10^-2 = 1/2*0.2*vi^2   => vi = 0.8m/s

initial displacement , xi

PEi = 1/2kxi^2

0.130 = 1/2*10.5*xi^2   => xi = 0.16m

= sqrt(k/m) = sqrt(10.5/0.2) = 7.25 rad/s

Use formula,

v = Acos(t + )

At t=0s

v = Acos

= cos^-1(v/A) = cos^-1[(0.8)/(0.192*7.25)] = 54.92 deg

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