Pendulum with Changing Mass

## Question:

How would the energy of a pendulum (performing S.H.M.) change when an external mass is dropped onto the bob at equilibrium position and at the extreme? I think if the mass is dropped at equilibrium, the K.E. will increases while the P.E. has no change just after it touches the bob. Total energy will increase. At extreme, the P.E. will increase while K.E. has no change. Is this correct? Will the loss in P.E. of that mass be converted to extra energy on the pendulum system? Why should it be better to illustrate this energy change example in a horizontal block-spring system?

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## Answer:

We need to think clearly about how this attaching additional mass to the pendulum bob takes place. If by "dropping a mass on the pendulum bob" we mean only that the mass of the bob increases because we arrange the "drop" such that its velocity matches that of the bob at the time of the contact, then we have one situation. If we are to consider the transfer of momentum from the incoming mass to the pendulum bob, that is a different situation.

If the incoming mass is dropped vertically downward so that it hits the bob at the bottom of its swing and sticks to it, the momentum of the incoming mass will be all in the downward direction, and the constraint connecting the bob to the pivot point will transfer all this momentum to the pivot support, effectively contributing zero momentum to the combined mass. This means that the change in momentum required to accelerate the new mass up to speed, must come from the initial momentum of the pendulum bob. The combined mass will move forward more slowly than the bob was originally traveling. The velocity of the combined mass will be reduced in proportion to the increase in mass.

The kinetic energy of the combined mass increases in proportion to the incoming mass and decreases in proportion to the square of the change in velocity so there will be a net loss of kinetic energy as a result of the perfectly inelastic collision between the bob and the incoming mass.

If we repeat the process when the pendulum is at an extreme of its motion, again throwing the incoming mass at the bob such that it strikes the bob from a direction parallel to the constraint between the pendulum bob and pivot, the incoming momentum is canceled by the constraint. The pendulum bob is at zero velocity at its extreme so the combined mass starts out on the next swing with zero velocity. The potential energy of the combine mass will be higher than it was before the bob took on the additional mass, by an amount proportional to the new mass. All of this potential energy will be converted to kinetic energy at the bottom of the swing, but since the combined mass has increased, the velocity at the bottom of the swing will be the same as it was for the lighter pendulum bob. In effect in this case, the amplitude and period of the pendulum remain the same.

If the incoming mass is added to the bob with any component of momentum either aiding or opposing the motion of the pendulum, the effects described will be altered by the amount of that momentum transfer.

If we were to conduct this experiment using a block sliding horizontally on frictionless rails with a spring as the restoring force, it would be easier to envision how the additional weight might be added. Also the nonlinear effects of the pendulum restoring force could be avoided.