Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth.

Answer 1

This is a little nebulous because it doesn't say what they want the energy to be calculated in terms of but we'll take a stab at it.

Assume the pendulum swings to angle θ, has a mass m, and a length l:

Using the triangle on the left, we can say sin(θ/2) = x/l or x = lsin(θ/2)

Using the bottom triangle, we know know the hypotenuse which is 2x or 2lsin(θ/2).

So we can say sin(θ/2) = h/(2lsin(θ/2)) so h = 2lsin²(θ/2)

That means the potential energy (gravitational potential energy) is mgh = 2mglsin²(θ/2) and since there is no velocity at the top of the swing, that is the total energy.

You could also do it this way:

We could say cos(θ) = z/l or z = lcos(θ)

Then h = l - lcos(θ) or l(1 - cos(θ)). Then the potential energy (and total mechanical energy = mgl(1 - cos(θ))

Either of these turns out to be the same and is acceptable.