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A mass m hangs in the presence of gravity on the end of a spring fastened...

  1. A mass m hangs in the presence of gravity on the end of a spring fastened at the top. The spring is initially outstretched. The mass is released at t = 0 with velocity = 0. Solve the differential equation that results from applying Newton’s second law [Hint: you should obtain an inhomogeneous second order differential equation where the homogeneous equation is just simple harmonic oscillator. Note the spring is under the action of two forces: the restoring force F = -kx and the gravitation force F= mg. So you need to have the net force equal to ma]
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Answer #1

Due to mg force , there will be an extension of spring by amoumt

L = mg/k

Now if we set origin of y-axis at L= mg/k

then equation of motion becomes homogeneous

, where ,

is solution of this homogenous equation

Replacing in our homogenous equation

Thus is a solution of HDE provided

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