Homework Problem

TQ 3-3

At its greatest height, the pendulum has zero kinetic energy.  Therefore, at its greatest height the energy is all mechanical.  At the bottom of its path, all the potential energy has been converted into kinetic energy.  Therefore, the kinetic energy will be greatest at the pendulum's lowest point.

Maximum potential energy is mgh_i, where h_i (the initial height) is measured from the lowest point the pendulum reaches.

Neglecting energy losses due to friction and air resistance, the mechanical energy (the total of the kinetic energy and potential energy) will not change at any position.

Then,

    PE_max = KE + PE

From the problem statement

    PE = KE

resulting in

    PE_max = 2PE

or

    PE = (1/2) PE_max

and

    mgh = (1/2)mgh_i

so

    h = (1/2)h_i

(half-way back up from lowest point)