Question

A 22.0 kg child plays on a swing having support ropes that are 1.90 m long. A friend pulls her back until the ropes are 45.0 ∘ from the vertical and releases her from rest. What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing?

Answer 1

Answer:

120 J

Explanation:

Given:

Mass of the child (m) = 22.0 kg

Length of the rope (L) = 1.90 m

Angle made by the rope (x) = 45.0°

Consider the diagram below for the given scenario.

Point A represents the bottom position, point B represents the maximum height reached, OA or OB is the length of rope.

OA = OB = L = 1.90 m

∠ AOB = x = 45.0°

Now, the maximum height reached by the child is given as:

$h=L-L\cos x\\\\h=L(1-\cos x)\\\\h=1.90\ m(1-\cos (45.0))\\\\h=0.556\ m$

So, as the child is pulled to a height 'h', the child's potential energy is increased.

Therefore, the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing is equal to the increase in the gravitational potential energy of the child.

The increase in gravitational potential energy is given as:

$\Delta U=mgh\\\\\Delta U=(22.0\ kg)(9.8\ m/s^2)(0.556\ m)\\\\\Delta U=119.9\approx 120\ J$

Therefore, the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing is 120 J.