advanced pendulum problem?

Question

A ball, attached to a string of length L = 107.0 cm, is released from a horizontal position, and when the string is vertical it hits a fixed peg at point P, located a distance d = 79.2 cm below the string's pivot point. The ball then swings around the peg, following a new (smaller) circle.
How fast will it be going when it reaches the highest point in its swing after hitting the peg?

Answer 1

A pendulum at its peak, no matter the pendulum, always has zero speed and max acceleration. Well, that's assuming it doesn't have enough swing to go all the way around the peg.

Now, to do it, calculate the potentiel energy of your pendulum in its initial state. Once it hits the peg, it'll have the same energy, but as kinetic energy. Now, it should have enough speed to go all the way around. The speed at the top will be the speed corresponding to the kinetic energy calculated earlier minus the potential energy over the peg.

Answer 2

You can solve this using SUVAT equations for motion, or conservation of energy. The latter is probably easier.
When an object is in a gravity well, such as the earth, it has a certain amount of potential energy.
In fact, potential energy = mgh, where m is the mass, g is acceleration due to gravity, and h is the height at which the object resides.
When the object descends, that energy is increasingly converted into kinetic energy.
Kinetic Energy = 1/2 mv^2, where m is the mass of the object, v is its velocity.
So, the pendulum effectively falls a vertical distance of 1.07m, hits the peg, then ascends again. It ascends a distance equal to 2(1.07 - 0.792) = 0.556 m (twice the radius of the circle it makes around the peg- draw a diagram if you have trouble following this).
So, overall it has gone from rest, descended 1.07m, then ascended 0.556 m, so in total, it has descended 1.07-0.556 = 0.514 m
So, if we call its starting height 'h(1)', and its final height 'h(2)', then we get a value for the change in its potential energy as a result of falling, equal to:
mgh(1) - mgh(2) = mg(h(1) - h(2)).
So, by substituting in the numbers (g = 9.8):
change in energy = m x 9.8 x (0.514) = 5.037 x m

Now, this energy has been converted to kinetic energy, = 1/2mv^2

Therefore, 1/2mv^2 = 5.037m.
Cancel 'm':
1/2v^2 = 5.037
Rearrange: v^2 = 10.074, v = (10.074)^1/2 = 3.17 m/s
Follow the method, but double-check the workings, as I have been known to make mistakes.
The principle behind this is that conservation of energy applies just as much to a falling pendulum as to a free-falling object, so you gain in kinetic energy and hence speed, is equal to the loss in potential energy as a result of descent.

Answer 3

No matter what kind of pendulum, the speed is zero because as the pendulum reaches its maximum point, it stops in midair and then changes its direction.