A child with a weight of 140 N swings on a playground swing attached to 2.10 m long chains. What is the gravitational potential energy (in J) of the child-earth system relative to the child's lowest position at the following times?
a. when the ropes are horizontal
b. when the ropes make a 25.0° angle with the vertical
c. when the child is at the bottom of the circular arc

a). Gravitational Potential energy= 294 Joules

b). when the ropes make a 25.0°

Gravitational Potential energy= 27.55 Joules

C).Gravitational Potential energy= 0 Joules

Explanation:

GPE (gravitational potential energy ) is always relative to some zero reference point.  For swings, it makes sense to set GPE = 0 at the bottom of the swing's arc where theta = 0 deg relative to the vertical.

In general GPE (gravitational potential energy ) = mgh;

where mg = 140 N is the weight of the child with mass m,

g is the gravity field, and

h is the height above the zero reference.

In general, from ordinary trig, we have h = L(1 - cos(theta))

where L = 2.1 m is the length of each rope.  

Gravitational Potential energy = mgh

Gravitational Potential energy

= 140 × L(1 - cos(90))

= 140 × 2.10(1 - 0)

= 294joules

b)  GPE = 140*2.1(1 - cos(25))

Gravitational Potential energy

= 140 × L(1 - cos(25))

= 140 × 2.10(1 - 0.9063)

= 140 × 2.10(0.0937)

= 140 × 0.19677

=27.55 joules

c)

GPE = 0  or

= 140*2.1(1 - cos(0))

= 0

at right angles to each other