An ideal simple pendulum consists of a point mass (bob) suspended from a fixed point by a massless, inextensible string of length L. When displaced from its equilibrium position and released, it oscillates back and forth under the restoring force component of gravity. For small angular displacements (less than 10°), the motion is simple harmonic.

The restoring force on the bob at displacement angle θ is F = −mg sinθ ≈ −mgθ (for small θ). This gives the SHM equation: d²θ/dt² = −(g/L)θ, where the angular frequency ω = √(g/L). The time period follows directly.

Key: time period depends only on length L and local gravity g. It does NOT depend on mass of bob, amplitude (for small oscillations), or material of bob. This property — called isochronism — makes pendulums excellent timekeepers.

In a real pendulum, the bob has finite size. The effective length L is the distance from the pivot point to the centre of mass of the bob, not just the string length. If the string length is l and the bob radius is r, then effective length L = l + r. This is what Savitha is trying to determine experimentally in this question.

Experimentally, the time period is measured for many oscillations (say 30 or 50) to reduce the error from starting/stopping the stopwatch. The time per oscillation T = total time / number of oscillations is then substituted in L = gT²/4π².

A second's pendulum has a time period of exactly 2 seconds (1 second for each half-oscillation — tick and tock). Its length at a location where g = 9.8 m/s²:

This is why grandfather clocks with a 1-metre pendulum tick every second. The second's pendulum length varies slightly with location because g varies with latitude and altitude.

The total mechanical energy of a pendulum is conserved (ignoring air resistance). At the mean position (bottom), all energy is kinetic: KE = ½mv²_max. At the extreme positions, all energy is potential: PE = mgL(1−cosθ₀) where θ₀ is the amplitude angle. For small angles: PE ≈ ½mgLθ₀² = ½mω²A² where A is the arc amplitude.

A physical pendulum is a rigid body oscillating about a pivot point that is NOT at its centre of mass. Its time period is T = 2π√(I/mgh) where I is moment of inertia about the pivot, m is total mass, and h is distance from pivot to centre of mass. A simple pendulum is a special case of physical pendulum where all mass is concentrated at one point at distance L from pivot.

In reality, a pendulum loses energy to air resistance and the amplitude gradually decreases — this is damped oscillation. The frequency of damped oscillation is slightly less than the natural frequency. In forced oscillation, an external periodic force is applied. When the external frequency equals the natural frequency, resonance occurs — amplitude becomes maximum. Resonance is used in radio tuning, musical instruments, and MRI machines.