Objective
The goal of this experiment is to show that energy is conserved when an object moves on a roller-coaster track. The work done to set the ball at the top of the track is converted from Potential Energy at the start to a combination of Potential and Kinetic Energy as it moves down.

Equipment
Roller-coaster track, steel ball, stand, meter stick, and photogate & timer interface.

Definition
The conservation of energy states that if you calculate the Total Energy_initial at the beginning, this will be exactly equal to the Total Energy_final at the end. In this experiment, you'll be sending a ball down a roller-coaster shaped track and calculating the total energy at every point along the path, or trajectory.

Total (mechanical) energy, in this case, consists simply of the Potential Energy and the Kinetic Energy. Potential energy measures the amount of energy something has because of its position in space; in the case of gravitational potential energy, this is the amount of work something can do if it falls under the influence of gravity. Kinetic Energy, on the other hand, measures the amount of energy something has due to its motion, or speed. So the Total Energy is simply the sum of the Kinetic and Potential energies; this is symbolized as: (Total Energy) = (Potential Energy) + (Kinetic Energy)

Kinetic Energy is defined as: KE = ½ mv², in which m is the mass of the ball and v is the velocity, or speed, of the ball [→ (Kinetic Energy) = ½ (mass) x (velocity)²]. Potential Energy, on the other hand, is defined as: PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity (measured in units of m/s²), and h is the height of the ball [→ (Potential Energy) = (mass) x (acceleration_due_to_gravity) x (height)]. The Total Energy, therefore, is symbolized as: E = PE + KE = mgh + ½ mv². In this lab, you'll calculate the Total Energy, E at various points along the path and see if total energy truly is conserved (i.e., if the number calculated for the total energy is the same at every point along the path).

Procedure

1. Setup the roller-coaster track as shown below. The mass of the ball is 28-grams (make sure convert this to kg) and its diameter is 1.89-cm (= 0.0189-meters). Make sure you put the threaded knob of the roller coaster knob into the 5^th hole from the bottom.

   

2. The plan is to measure the velocity (of the ball at various points down the track using the photogate. The measurement will be taken every 15-cm (using one photogate at each 15-cm increment). Measure the height of the ball at the starting point and also measure the height at each position of the clamp. Measure from the position of the light beam in the clamp. Make sure the clamp is flat against the bottom of the track so that when the ball passes through it, the diameter of the ball will pass across the beam.

   

3. Set the timer to interval mode so that it will record the time it takes for (the diameter of) the ball to pass through the photogate clamp.
   • The velocity at that point is given by:

     
     	         Δd        DIAMETERball                 0.0189-m
     	vball = ---- = -------------------- = ----------------------------------- [m/s]
     	         Δt     time_through_clamp     time_through_EACH_clamp (in secs)
     	

   • The Potential Energy (PE)_ball = mgh [Joules], where h is the height, g is the acceleration due to gravity (g = 9.8m/s²), and m is the mass of the ball.
   • The Kinetic Energy (KE)_ball = ½mv² [Joules], where v is the velocity and m is the mass of the ball.
4. Calculate the Total Energy by adding the PE and the KE. Now do this calculation for each position of the clamp down the track.
5. Plot the PE, the KE, and the Total Energy as shown below:

   

   Remember that, at the start (when the ball is at rest), the KE is zero. Notice that the total energy is conserved with about 15% being lost by the time the ball gets to the bottom of the track.

Questions

1. What is causing the loss of energy as the ball moves down the track?
2. If we used a ball with a smaller mass, what would the data look like for a) velocity and b)the Total Energy?
3. If there is sufficient time, check this out with the lighter ball.

Notes

1. Make sure that you start the ball at rest and don't push it along.
2. Use the clamps as following:
   1. Position photogate/clamp A at about the 0-cm mark (note the exact position) and clamp B at the 15-cm mark. Take your data.
   2. Now move clamp A to the 30-cm mark and clamp B to the 45-cm mark and record the next set of data.
   3. Continue in this fashion until you reach the end of the track.
3. You only need to record the height and time for each photogate/clamp position (the rest is just calculations).