Given the length of a pendulum (L), the period of the pendulum (T) is inversely proportional to the square root of the mass (m). This means that as the mass of the pendulum increases, the period decreases. Mathematically, this relationship can be expressed as:
T = 2π√(L/g)
where g is the acceleration due to gravity.
This inverse relationship can be explained by the principle of conservation of energy. As the pendulum swings back and forth, the potential energy it gains at the top of its swing is converted into kinetic energy as it falls. The period of the pendulum is determined by the time it takes for the energy to transfer back and forth between potential and kinetic forms.
The mass of the pendulum affects the kinetic energy of the pendulum, but it does not affect the potential energy. This is because the potential energy depends only on the height of the pendulum bob, which is independent of the mass of the bob. As the mass of the pendulum increases, the kinetic energy of the pendulum also increases. However, the kinetic energy of the pendulum is proportional to the square of the mass, while the period of the pendulum is inversely proportional to the square root of the mass. Therefore, as the mass of the pendulum increases, the kinetic energy increases by a greater factor than the period decreases, and the overall effect is that the period decreases.
In practice, the effect of mass on the period of a pendulum is very small. For example, a pendulum with a mass of 1 kg will have a period of about 2.0 seconds, while a pendulum with a mass of 2 kg will have a period of about 1.9 seconds. This is a difference of only 0.1 seconds. Therefore, for most purposes, the mass of the pendulum can be considered to be a constant and the period can be approximated as being independent of the mass.