Question

A 40.0 kg child is in a swing that is attached to ropes 1.60 m long. The acceleration of gravity is 9.81 m/s. Find the gravitational potential energy associated with the child relative to the child’s lowest position under the following conditions:
a. when the ropes are horizontal. Answer in units of J.
b. when the ropes make a 36.0◦ angle with the vertical. Answer in units of J.
c. at the bottom of the circular arc. Answer in units of J.

Answer 1

Answer:

a) 627.84 Joules

b) 117.72 Joules

c) 1255.68 Joules

Explanation:

(See figure 1)

The gravitational potential energy relative to the child’s lowest position is:

$U_{g}=mgh$ (1)

with h the vertical distance of the swing from the lowest position, m the mass of the child and g the acceleration of gravity.

a) When the ropes are horizontal, the swing is at 1.60 m from the lowest position, so by (1):

$U_{g}=(40)(9.81)(1.60)=627.84\,J$

b) When the ropes make a 36.0◦ angle with the vertical, the swing is at a distance d from the lowest position, we should use trigonometric relations to find that distance. By figure 2 we have a right triangle with adjacent side $1.60 - d$ and hypotenuse 1.60, so using the trigonometric relation $\cos(36)=\frac{(1.60-d)}{1.60}$ we can solve for d:

$d=1.60-1.60*\cos(36)\approx0.30\,m$

Using d on (1):

$U_{g}=(40)(9.81)(0.30)=117.72\,J$

c) Note that at the bottom of the circular arc the distance of the swing relative to the lowest position is two times the length of the rope, so h=3.20 m, using this on (1):

$U_{g}=(40)(9.81)(3.20)=1255.68\,J$