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:
GPEA = KEB
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:
Solving for vo 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 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:
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?)
Calculate for each trial (show a sample calculation) and determine the probable uncertainty in this value.
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.