Question

A 38.0 kg child is in a swing that is attached to ropes 1.50 long. The acceleration of gravity is 9.81 m/s^2 . Find the gravitational potential energy associated with the child relative to the child's lowest position under the following condition: when the rope makes a 34.0 degree angle with the vertical.

Answer 1

Answer on Question#37807, Physics, Mechanics

Question:

A 38.0 kg child is in a swing that is attached to ropes 1.50 long. The acceleration of gravity is 9.81 m/s^2. Find the gravitational potential energy associated with the child relative to the child's lowest position under the following condition: when the rope makes a 34.0 degree angle with the vertical.

Answer:

The gravitational potential energy of the child relative to the child's lowest position is

$$U = m g h$$

where $m$ – is the mass of the child, $g$ – acceleration of gravity, $h$ – the height of the child relative to the child's lowest position.

From the similarity of triangles that appears from the child's displacement from vertical we have

$$h = b \cos \alpha$$

where $\alpha$ is angle that rope makes with the vertical and $b$ is

$$b = \frac{l}{\cos \alpha} - l$$

where $l$ the length of the rope.

So we have

$$h = l (1 - \cos \alpha)$$

and the potential energy is

$$U = m g l (1 - \cos \alpha)$$ $$U = 38.0 \cdot 9.81 \cdot 1.50 \cdot (1 - \cos 34.0{}^{\circ}) = 95.60 \, J$$

The answer is: $U = 95.60 \, J$