Pendulums

The period of oscillation of a simple pendulum moving down an inclined plane is given by the formula: This formula accounts for the motion of the pendulum on an inclined plane where the effective gravitational force is .

The potential energy of the pendulum at the top (point A) is converted into kinetic energy at the bottom (point B), with some energy lost due to air resistance.
The initial potential energy is given by: where is the length of the pendulum, and is the vertical height. At the lowest point, the kinetic energy is: The energy dissipated due to air resistance is 5% of the total energy, so the remaining energy is 95%.
Therefore: Substituting , we get: Simplifying:
Thus, the correct answer is (c).

The vibrational energy of a simple pendulum is given by: where is the height change corresponding to the amplitude of oscillation.
Since the vibrational energies are the same for both pendulums, the height change and thus the amplitude for the pendulum with the shorter length (pendulum ) will be smaller than the amplitude for the pendulum with the longer length (pendulum ).
Thus, the correct answer is (a).