Question

Hi! I know the answer is 1/4 but I would like to know the explanation for this. I would appreciate an in depth response if possible as this kind of question will be on my midterm. Thanks so much!

A mass attached to a spring is executing simple harmonic motion with an amplitude of 20 cm. When the mass is at a distance of 10 cm from the equilibrium position what fraction of its total energy is the kinetic energy of the mass?

Answer 1

First of all I want to let you know that

the spring energy at that point is (1/4) of the total energy

And

The kinetic energy at that point is (3/4) of the total energy.

Le me derive each of them

===========

Max spring energy = 0.5 k A^2

A(amplitude) = 20 cm

x ( amplitude) = 10 cm

It means

x/ A = 10/20

x = A/2

NOW

Spring energy at that instant = 0.5 k x^2 = 0.5 k (A/2)^2 = 0.5 k A^2 / 4

From (i)

Spring energy at that instant = max spring energy / 4

======

For kinetic energy

Using conservation of energy

Max spring energy = kinetic energy + spring energy at that instant

KE = 0.5 k A^2 - 0.5 k A^2/4

KE = (3/4) 0.5 k A^2

KE = (3/4)* max spring energy stored in spring

=======

Plz do comment if any doubt persist in the solution, will reply for sure.. Goodluck