Question

A child of mass M is swinging on a swing set. The ropes attaching the swing to the top bar have length L. Find the gravitational potential energy of the child relative to her lowest position when the ropes are (a) hanging straight down; (b) exactly horizontal; (c) at an angle ϴ from the vertical; and (d) at an angle phi from the horizontal.

Answer 1

(a) 0

(b) 10ML

(c)
`10ML(1 - cos(\theta))`

(d)
`10ML(1 + sin(\phi))`

Step-by-step explanation:

(a) When hanging straight down. The child is at the lowest position. His potential energy with respect to this point would also be 0.

(b) Since the rope has length L m. When the rope is horizontal, he is at L (m) high with respect to the lowest swinging position. His potential energy with respect to this point should be

`E_h = mgh = 10ML`

where g = 10m/s2 is the gravitational acceleration.

(c) At angle
`\theta` from the vertical. Vertically speaking, the child should be at a distance of
`Lcos(\theta)` to the swinging point, and a vertical distance of
`L - Lcos(\theta)` to the lowest position. His potential energy to this point would be:

`E_(\theta) = mgh = 10M(L - Lcos(\theta)) = 10ML(1 - cos(\theta))`

(d) at angle
`\phi` from the horizontal. Suppose he is higher than the horizontal line. This would mean he's at a vertical distance of
`Lsin(\phi)` from the swinging point and higher than it. Therefore his vertical distance to the lowest point is
`L + Lsin(\phi) = L(1 + sin(\phi))`

His potential energy to his point would be:

`E_(\phi) = mgh = 10ML(1 + sin(\phi))`