The work done to pull the sister back on the swing is equal to the increase in potential energy of the sister:
[tex]W= \Delta U = mg \Delta h[/tex] (1)
where m is the sister's mass, g is the gravitational acceleration and [tex]\Delta h[/tex] is the increase in altitude of the sister with respect to its initial position.
By calling [tex]\theta[/tex] the angle of the chain with respect to the vertical, the increase in altitude is given by
[tex]\Delta h = L - L \cos \theta = L(1 - \cos \theta)[/tex] (2)
where L is the length of the chain.
Putting (2) inside (1), we find
[tex]W= m g L (1 - \cos \theta)[/tex]
from which we can find the mass of the sister:
[tex]m = \frac{W}{g L (1 - \cos \theta)} = \frac{120 J}{(9.81 m/s^2)(5.10 m)(1- \cos 32.0^{\circ})} =15.8 kg[/tex]
