energy

N otes on Chapter 7 B
pp 69 - 72 of your lab book

Potential energy is stored energy
Gravitational potential energy is energy which an object has due to its position in a gravitational field (the space around any mass).
Kinetic energy is energy which an object has due to its movement (velocity)

Potential energy is not immediately useful. In order to get energy that is useful, we need to convert potential energy into kinetic energy. If you held a ball at a height h above the ground and then let go of it, you would see the ball drop. The ball, which had no velocity at the moment you let go, has velocity when it reaches the ground. So initially the ball has no kinetic energy. When it hits the ground it now has kinetic energy. Did the energy come from nowhere? Of course not! The kinetic energy came from the potential energy the ball had when it was at height h.

Conservation of energy: the total energy of a system before = the total energy of a system after
Total energy = potential energy + kinetic energy

The conservation of energy tells you how much potential energy got converted into kinetic energy. This conservation law is true provided no work was done. Work can be an energy loss in your system due to friction, deformations as the object collides with something else, or it can be an energy gain to your system such as wind. (But remember the total energy of the entire universe is always conserved, so although your system is losing energy, something else has to be gaining energy!)

The equation for gravitational potential energy (PE) = mgh, where m = mass of the object, g = the gravitational acceleration, and h = height
The equation for kinetic energy (KE) = 1/2 mv², where v = velocity.

Notice that the conservation formula has no information in it about what path the object took or what time means "before" and "after". You could look at a ball which starts at height h, goes through 12 loop-the-loops, spins around in a corkscrew, and an hour later hits the ground, and you could say "before" was at height h, and "after" was an hour later when the ball hits the ground. The conservation law would still be true! So in our lab we will use a pendulum. We will start it at height h and see what kinetic energy it has at the ground. It doesn't matter that the pendulum is swinging in an arc.!

In the picture above, the pendulum is released from rest at a height h. We want to measure the energy when the pendulum is at the lowest point in its swing, which we will say is ground (height = 0 meters). Our conservation law is:

mgh_i + 1/2 mv_i² = mgh_f + 1/2 mv_f²
mgh + 1/2 m(0 m/s)²= mg (0 meters) + 1/2 mv²
mgh = 1/2 mv²

How do we measure the velocity of the pendulum? In order to do this, we place a ball in the path of the pendulum. The ball is of the same mass as the pendulum bob, so the pendulum will give all its kinetic energy to the ball. We can measure the velocity of the ball because the ball is a freely falling body, and we know the equations of motion for that! If we know the distance it travels before it lands and the height at which it started, then we can determine the velocity at the beginning of the path, which is the same as the velocity of the pendulum at the lowest point!

Putting all these facts together, we can come up with an equation for D, the distance the ball travels, in terms of L, the height that the ball is at relative to the floor, and h, the height of the pendulum bob. (See figure 7.4) We will derive this in class.

D² = 4Lh

Of course, energy is lost to friction and to the collision. You can calculate the theoretical D from your measurements of L and h, and that calculated D will always be greater than the D which you observe.

Trial	Height 1: Distance Measured ( )	Height 2: Distance Measured ( )	Height 3: Distance Measured ( )
1
2
3
4
5
average +/-SDM


bad	bad	good