A190 n child is in a swing that is attached to ropes 2.10 m long. find the gravitational potential energy of the child-earth system relative to the child's lowest position at the following times.
(a) when the ropes are horizontal the correct answer is not zero.
(b) when the ropes make a 34.0° angle with the vertical.
(c) when the child is at the bottom of the circular arc

a) Gravitational potential energy = 399 J

b) Gravitational potential energy = 66.5 J

c) Gravitational potential energy = 0 J

Explanation:

Hi there!

Please, see the attached figure for a better understanding of the problem.

a) When the ropes are horizontal, the height of the child, relative to the child's lowest position, is 2.10 m (see figure).

The gravitational potential energy is calculated as follows:

PE = mgh

Where:

PE = potential energy.

mg = weight of the child

h = height.

Then when the ropes are horizontal, the potential energy will be:

PE = 190 N · 2.10 m = 399 J

b) When the ropes make a 34.0° with the vertical, the height of the child is 2.10 m minus x (see figure). To find x, we can use trigonometry of right triangles:

cos angle = adjacent side / hypotenuse

cos 34.0° = x / 2.10 m

x = 2.10 m · cos 34.0° = 1.75 m

Then, the height of the child relative to the lowest position is

(2.10 m - 1.75 m) = 0.35 m

Therefore, the gravitational potential energy will be:

PE = 190 N · 0.35 m

PE = 66.5 J

c) When the child is at the bottom of the circular arc the height is zero (the child is at the lowest position), then, the gravitational potential energy will be zero.