Simple Pendulum
T = 2π√(L/g) — animated swing.
Key Notes
Simple pendulum: a point mass m suspended from a massless string of length L, swinging under gravity.
For small angles (θ < ~15°), sin θ ≈ θ ⇒ motion is approximately SHM.
Equation of motion (small angle): θ̈ + (g/L)·θ = 0 ⇒ ω = √(g/L).
Period T = 2π·√(L/g). Independent of mass m and amplitude (for small angles).
T depends on L and g — does NOT depend on mass m of the bob.
Long pendulums swing slowly; short ones swing fast.
At higher altitudes or at the equator: g lower ⇒ T longer.
Foucault pendulum: long pendulum demonstrates Earth's rotation as its swing plane appears to rotate.
Formulas
Equation of motion (small angle)
Linear approximation valid for small θ.
Angular frequency
Independent of mass and amplitude (small).
Period
Doubling L ⇒ T × √2.
Large-amplitude correction (1st order)
Period grows with amplitude beyond ~15°.
Important Points
T is INDEPENDENT of mass m and (to first order) of amplitude.
T depends only on L and g — used historically to measure g.
Pendulum at top of mountain runs SLOW (g smaller). Pendulum on Moon would have T ~ 2.45× Earth value (g ≈ 1.6 m/s²).
Pendulum clocks: precise T relies on uniform L and g.
Beyond ~15°, amplitude affects T noticeably (anharmonic correction).
Foucault pendulum: swing plane rotates due to Earth's rotation. Period of rotation ~ T_Foucault = 24 h / sin(latitude).
Simple Pendulum 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.