The equation of motion of an undamped pendulum

[Yeep]

The equation of motion of an undamped pendulum (i.e., there is no friction) is
mLy'' = −mg sin y.

The pendulum is made by attaching a weight of mass m to a very light andrigid rod of length L mounted on a pivot so that the system can swing in avertical plane. Here y denotes the angle that the pendulum makes with thevertical (equilibrium) position and g is the acceleration of gravity. Note that−π/2 < y < π/2. The restoring force, due to gravity, is fre(y) = −mg sin y.(a) Determine the pendulum’s potential energy p(y) and sketch its graph.(b) Determine the pendulum’s total mechanical energy E(t).(c) State the principle of conservation of total energy and show why it holds forthe pendulum.

Honestly have no idea where to begin to find the potential energy, let alone sketch a graph of it. Can anyone give me a push in the right direction?

 

[HallsofIvy]

The "independent variable", t, does not appear explicitly, and there is a regular method for such equations: let v= y'. Then y''= dv/dt= (dv/dy)(dy/dt)= v(dv/y)and the equation becomes mLv(dv/dy)= -mg sin(y). That can be written mvdv= -m(g/L) sin(y)dy. Integrating both sides (1/2)mv^2= m(g/L) cos(y)+ C so (1/2)mv^2- m(g/L) cos(y)= C, a constant.

Do you see that one of those two terms on the left is kinetic energy and the other is potential energy?