Question

How do i do this simple harmonic motion?

When the displacement in SHM is equal to `1/6` of the amplitude `X(m)`, what fraction of the total energy is
(a) kinetic energy and
(b) potential energy?
(c) At what displacement, in terms of the amplitude, is the energy of the system half kinetic energy and half potential energy? (Give the ratio of the answer to the amplitude)

Answer 1

See below

Explanation:

For a system with linear restorative force `F = -kx`, which is the hallmark of SHM:

• Total Energy `E = T + U`

• `T = 1/2 m dot x^2, qquad U = 1/2 k x^2`

Now:

`x = X implies T = 0 , qquad implies E = U_(max) = 1/2 k X^2`

So:

`T + U = 1/2 k X^2`

(a)

For `x = X/6`:

`T + 1/2 k (X/6)^2 = 1/2 k X^2 implies T = 1/2 k X^2 * 35/36 = 35/36E`

`color(red)(implies T/E = 35/36 )`

(b)

Follows that:

`color(red)(implies U/E = 1/36 )`

(c)

If `T = U`, then:

`2U = E`

`2* 1/2 k x^2 = 1/2 k X^2`

`color(red)(implies x = X/sqrt2)`