a) At the highest point, all the potential energy is in the form of gravitational potential energy, which is given by:
U = mgh
where m is the mass of the child, g is the acceleration due to gravity, and h is the height above the lowest point (i.e., the bottom of the swing). At the highest point, h = 2.20 m, so:
U_top = mgh = (25 kg)(9.81 m/s^2)(2.20 m) = 544.5 J
At the bottom of the swing, all the potential energy has been converted to kinetic energy, which is given by:
K = (1/2)mv^2
where v is the velocity of the child at the bottom of the swing. Since the swing is released from rest, the initial velocity is zero. Therefore, the kinetic energy at the bottom of the swing is equal to the potential energy at the top of the swing:
K_bottom = U_top = 544.5 J
b) Setting the potential energy at the top of the swing equal to the kinetic energy at the bottom of the swing, we can solve for the velocity:
K_bottom = (1/2)mv^2
544.5 J = (1/2)(25 kg)v^2
v = sqrt(2*544.5 J / 25 kg) = 9.89 m/s
Therefore, the child will be moving at a speed of approximately 9.89 m/s at the bottom of the swing.
c) The work done by the tension in the ropes is equal to the change in kinetic energy of the child as she swings from the initial position to the bottom. Since the child starts from rest, the initial kinetic energy is zero. Therefore, the work done by the tension is equal to the final kinetic energy at the bottom of the swing:
W = K_bottom = (1/2)mv^2 = (1/2)(25 kg)(9.89 m/s)^2 = 1228.3 J
Therefore, the tension in the ropes does 1228.3 J of work as the child swings from the initial position to the bottom.