Motion under Gravity

1. Period (T): The time taken by the pendulum bob to complete one full oscillation (to and fro).
2. Angular displacement: The angle made between the string and the vertical axis when the bob is displaced to the maximum position.
3. Amplitude: The maximum displacement of the bob from the mean position (equilibrium).
4. Length of pendulum (l): The distance from the point of suspension to the center of gravity of the bob.

The period of a simple pendulum is given by the formula:

Derivation step-by-step:

Step 1: Start with the basic period formula

Step 2: Square both sides to eliminate the square root:

Step 3: Rearranging to express length (l):

If we plot a graph of l against T², the slope of the straight line will be:

And the y-intercept will be 0 since the equation is of the form $y = mx$.

1. When the bob is raised to point B (or C), it possesses potential energy (P.E.).
2. As it swings down to the lowest point O, the potential energy is converted into kinetic energy (K.E.).
3. At points B and C: All energy is potential energy (P.E.).
4. At point O: All energy is kinetic energy (K.E.).

By conservation of energy:

Energy loss in real-life: In a real pendulum, due to air resistance and internal friction, energy is gradually lost, and the amplitude decreases until the pendulum stops oscillating. However, in a vacuum, the pendulum would swing indefinitely since there's no energy loss.