# Pendulum hitting a peg/conservation of energy

## Homework Statement

A pendulum comprising a light string of length L and a small sphere swings in the vertical plane. the string hits a peg located a distance d below the point of suspension. (Diagram attached below).

a) Show that if the sphere is released from a height below the peg, it will return to this height after the string strikes the peg.

b) Show that if the pendulum is released from rest at the horizontal position (θ = 90°) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5.

## Homework Equations

Etotal = ΔK + ΔU = 0 for an isolated system

## The Attempt at a Solution

I've attempted a) so far, but I'm not sure if it's right so I haven't yet tried b).

For a), I assumed that the only "energy" acting on the system in both the initial and final state would be gravitational potential energy. (No kinetic energy because it's released from rest, and v = 0 at the point where the pendulum turns to swing back).

So this gives me:

Ugf - Ugi = 0
mg(Lsinθ - d) - mgh = 0 (where 'h' is my unknown final height)
h = Lsinθ - d

I'm not sure if my "Lsinθ - d" value is correct for the initial height... would appreciate if someone could check this and/or lead me in the right direction if I'm wrong!

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