Because the distance from the where the pendulum start ,it cover equal distance in both sides so ,equal time in one oscillation...

The Pendulum

A pendulum is a weight suspended from a fixed point that swings back and forth under the force of gravity. It consists of a mass, called the bob, attached to a string or rod of a fixed length. When the pendulum is displaced from its resting position and released, it swings back and forth in a regular pattern known as oscillation.

Oscillation of a Pendulum

The oscillation of a pendulum is characterized by two main properties: period and frequency. The period of a pendulum is the time it takes for one complete back and forth swing, while the frequency is the number of swings per unit of time.

Equal Times in One Oscillation

The correct answer to the question is option C: Equal times in one oscillation. This means that a pendulum of a given length takes the same amount of time to complete one full swing back and forth.

This property of a pendulum is known as isochronism, which was first discovered by Galileo Galilei in the late 16th century. Galileo found that regardless of the amplitude (the angle of displacement from the resting position), a pendulum of a given length will always take the same amount of time to complete one swing.

Explanation

To understand why a pendulum takes equal times in one oscillation, we need to consider the forces acting on it. When a pendulum is displaced from its resting position and released, it experiences two main forces: gravity and tension.

- Gravity: The weight of the bob pulls it downward, creating a restoring force that tries to bring the pendulum back to its equilibrium position.
- Tension: The tension in the string or rod pulls the bob towards the center, providing the necessary centripetal force to keep the pendulum swinging in a circular motion.

Simple Harmonic Motion

When the angle of displacement is small (less than 20 degrees), the motion of a pendulum is approximately simple harmonic motion (SHM). In SHM, the restoring force is directly proportional to the displacement but in the opposite direction. This means that as the pendulum swings to one side, the force of gravity tries to bring it back, and as it swings to the other side, the tension in the string pulls it back.

Period of a Pendulum

The period of a pendulum is determined by the length of the string or rod and the acceleration due to gravity. It can be calculated using 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.

Since the formula for the period does not depend on the amplitude or the initial displacement, the pendulum will always take the same amount of time to complete one swing, regardless of how far it swings.

Conclusion

In conclusion, a pendulum of a given length takes equal times in one oscillation. This is due to the isochronism property of a pendulum, where the period of oscillation is independent of the amplitude or initial displacement. This property was first discovered by Galileo Galilei and can be explained by the forces of gravity and tension acting on the pendulum. The period of a pendulum is determined by its length and the acceleration due to gravity, resulting in equal times for each swing.