AP Physics Lab

Conservation of Energy

in a Pendulum


Pendulum diagramSuppose that a pendulum bob of mass m is raised through a height h (point A in the diagram) and released from rest. At the lowest point of its path (point B in the diagram), suppose the mass is released and allowed to fall freely through a vertical distance y to the table, where it lands at point C as shown. Is energy conserved in this motion?

If point B is taken as the zero point of gravitational potential energy, the total mechanical energy of the pendulum at point A is its gravitational potential energy (since it is at rest. At point B, its total mechanical energy equals its kinetic energy (since its GPE = 0). Therefore, if mechanical energy is conserved:


If point A is a vertical distance h above point B, and if the mass has velocity vo when it reaches B, this means that:

mgh = (1/2)mv^2

Solving for vo gives:

vo = sqrt(2gh)

Now, suppose that at point B the mass is released from the pendulum and allowed to move as a projectile. The horizontal distance that it will move (at constant velocity vo) during the time t that it is in the air is:


But what is the value of t? In the vertical direction, suppose that the projectile falls a vertical distance y while moving from B to C. Then,


solving this for t gives:




which simplifies to:

x = sqrt(hy)

So, if mechanical energy is conserved in this situation, the three distances h, x, and y will have this relationship. (Isn't it interesting that x does not depend on m or g?)


pendulum apparatus

meter stick


  1. Set up the pendulum apparatus as your instructor directs.
  2. The vertical distance, y, will be constant for all of your trials, so measure it now. Have your lab partner measure y also, before disclosing your measurement, then compare and decide on the uncertainty in this measurement.
  3. Construct a data table to hold a value for h, x, and 2 times sqrt(hy)for each trial.
  4. How will you decide on an uncertainty for h and x? A little engineering at this point might help reduce this uncertainty, or at least pin it down.
  5. Run the experiment for as wide a range of h values as is practical. It would be wise to do some rough calculations of x and 2 times sqrt(hy)as you go - if there are large discrepancies you won't be able to do anything about them after you disassemble your apparatus!


Calculate 2 times sqrt(hy)for each trial (show a sample calculation) and determine the probable uncertainty in this value.


How does the value of x compare to 2 times sqrt(hy)? Was mechanical energy conserved for the motion? What are the major sources of error in this experiment? Justify your answers.

last update November 10, 2007 by Jerry L. Stanbrough