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?
Just reply when you have time! Thanks very very very
much!!!

## 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.