Final answer:
The period of a pendulum is independent of the mass it is holding due to the nature of the restoring force provided by gravity, which is proportional to the displacement, not the mass. The formula for the period of a pendulum, T = 2π√(L/g), confirms this, as the mass does not factor into the equation.
Step-by-step explanation:
The reason why the period of the pendulum motion does not depend on the number of dresses the rack is holding is due to the nature of restoring forces in pendulums. A simple pendulum, like the clothing rack described, oscillates due to the restoring force arising from gravity and this force acts proportionately to the displacement.
The period of a pendulum is determined by the formula T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. Notably, you can see that the mass (or number of dresses) does not appear in this equation, which is why the period is independent of the mass.
When we say that the period of a pendulum is independent of the mass, it assumes that the mass is distributed evenly along the length of the pendulum and doesn't affect its length. Since gravity acts as the restoring force and not the mass of the dresses, the swinging of the rack will have the same period regardless of how heavily loaded it is.
This principle is in line with the characteristics of harmonic motion, where the frequency of oscillation is independent of amplitude and, in this case, the mass attached to the pendulum.