Pendulum: Length vs Period
T ∝ √L plot.
Key Notes
Period of a simple pendulum is T = 2π√(L/g) — proportional to √L.
Doubling length increases period by √2 (~1.41×).
T² vs L is LINEAR with slope 4π²/g. Lab method to measure g.
Length L is measured from pivot to center of mass of bob.
For compound pendulum (extended body): T = 2π·√(I/Mgd), where I = moment of inertia about pivot, d = distance from pivot to COM.
Pendulum clock: typical L ≈ 1 m for T = 2 s. Grandfather clocks tune by adjusting L slightly.
Used in seismometers, accelerometers, time-standards before atomic clocks.
Temperature effects: warm air expands the pendulum rod ⇒ longer L ⇒ slower clock.
Formulas
Length-period relation
Linear in √L.
Inverted form for g measurement
Pendulum experiment classic.
Compound pendulum
I = moment of inertia about pivot; d = pivot-to-COM.
Important Points
T ∝ √L — exponent ½, not 1.
T² vs L gives a straight line through origin — slope = 4π²/g.
L is measured from pivot to bob's CENTER OF MASS (not to the bottom of the bob).
Pendulum clocks tune by ADJUSTING L: slightly longer ⇒ slower; slightly shorter ⇒ faster.
Compound pendulum has different T because I includes contributions from extended body, not just point mass.
Common pitfall: forgetting the square root. 4× length = 2× period (not 4×).
Pendulum: Length vs Period notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.