Question

derive an expression for the time period of a loaded spring

Answer 1

Answer:

The period of a mass m on a spring of spring constant k can be calculated as T=2π√mk T = 2 π m k .

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Answer 2

Answer:

You need to know the equation of motion. The force for the pendulum is given by F=−kx. Newtons equation tell you F=ma=mx¨. So you need to solve

mx¨=−kx.(1)

You know that the solution will be of oscillatory form. So you set x=Acos(2πt/T) and you want to obtain T. Plugging this ansatz into the equation (1), you obtain

−m(2π)2T2Acos(2πt/T)=−kAcos(2πt/T).

You see that the equation is fulfilled if

m(2π)2T2=k.

Solving for T, you obtain the result.