Simple Pendulum?

Question

The figure shows a simple pendulum of length L = 54 cm and mass m = 2.4 kg. It's bob is observed to have a speed of vo = 4.8 m/s when the cord makes an angle θo = 24°.

What is the speed of the bob when it is in its lowest position?

What is the least value that v0 must have if the cord is to swing up to a horizontal position?

Answer 1

Compute its total energy, E = PE+KE = mgh+mv^2/2; this will remain the same for any position
For the first question,
E(24deg) = E(0deg)
E(24deg) = mgh + mv^2/2,
where h = L sin(theta), v = 4.8
E(0deg) = mgh + mv^2/2,
where h = 0; thus mv^2/2 = E(24deg)
Solve for v.
For the second question,
E(24deg) = E(90deg)
E(90deg) = mgh + mv^2/2,
where h = L, v = 0; thus E(90deg) = mgL
E(24deg) = mgh + mv^2/2,
where h = Lsin(theta);
thus mv^2/2 = mgL(1-sin(theta))
Solve for v.