# Energy problem with pendulum

1. Mar 24, 2013

### Mangoes

1. The problem statement, all variables and given/known data

A simple pendulum has a length of 75 cm and a bob of mass 0.6 kg. When the string is at 30 degrees to the vertical, the bob has a speed of 2 m/s. What is the maximum speed of the bob?

3. The attempt at a solution

The main idea I had was that the velocity of the bob should be highest where there's the most kinetic energy and the least potential energy. The lowest point on the pendulum will be when it's stretched completely down and where it has the least potential energy. I marked the initial position as state 1 and the lowest point as state 2.

http://i.imgur.com/pmNOYbf.png

The tension force does no work on the bob since it always acts perpendicular to the displacement. This only leaves gravity doing work on the bob.

I chose state 2 to be y = 0. The vertical component of the pendulum's length in state one is 0.75sin30 = 0.375. By luck, the distance between the bob and pivot point is the same as the distance from the bob to y = 0.

In state 1,

$$K_1 = 0.5mv^2 = 0.5(0.6kg)(2m/s)^2 = 1.2J$$

$$U_1 = mgh = (0.6kg)(9.8 m/s^2)(0.375m) = 2.205 J$$

In state 2, the potential energy is zero by my choice of point of reference.

By the conservation of energy, the kinetic energy at this point must be equal to the sum of the potential and kinetic energy in state one.

$$K_2 = (1.2 + 2.205)J = 3.405 J$$

By the definition of kinetic energy,

$$3.405 J = 0.5(0.6kg)v^2$$
$$v = 3.36 m/s$$

Don't see how I'm going wrong here, but my answer isn't matching up. The answer given in my sheet is 2.44 m/s.

Where am I going wrong with this?

2. Mar 24, 2013

### bp_psy

Your sketch is wrong. The angle is to be taken from the vertical.