Discussion:

Suppose 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:

GPE_{A}= KE_{B}

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

Solving for v_{o} gives:

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 v_{o}) 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:

so:

which simplifies to:

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?)

Equipment:

pendulum apparatus |
meter stick |

Procedure:

- Set up the pendulum apparatus as your instructor directs.
- 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. - Construct a data table to hold a value for h, x, and for each trial.
- 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.
- 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 as you go - if there are large discrepancies you won't be able to do anything about them after you disassemble your apparatus!

Results:

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

Conclusions:

How does the value of x compare to ? 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