Simple Pendulum
Drag the bob to any angle and release. Adjust length, gravity, mass and damping. Watch live graphs and verify T = 2π√(L/g) with your own observations.
| # | Length L (m) | Mass m (kg) | Gravity g (m/s²) | Angle θ₀ (°) | Damping b | T measured (s) | T formula (s) | Frequency f (Hz) |
|---|---|---|---|---|---|---|---|---|
| No observations yet. Press "+ Record" to save current values. | ||||||||
Theory — Simple Pendulum
A simple pendulum consists of a point mass (bob) suspended from a fixed pivot by a massless, inextensible string. When displaced from its equilibrium and released, the bob oscillates under gravity in a periodic motion.
For small angles (θ < 15°), the motion approximates simple harmonic motion (SHM), where the restoring force is proportional to the displacement from equilibrium.
This gives sinusoidal oscillation: θ(t) = θ₀ cos(ω₀t), where ω₀ = √(g/L) is the natural angular frequency.
The Period Formula
The time period (T) is the time for one complete oscillation. For small angles it is given by:
Key observations from the formula:
- T increases with length L (longer = slower)
- T decreases as gravity g increases (stronger gravity = faster)
- T does NOT depend on mass of the bob
- T is independent of amplitude for small angles
- Doubling L increases T by a factor of √2 ≈ 1.41
Energy in a Pendulum
As the pendulum swings, energy continuously converts between kinetic and potential forms, but total mechanical energy is conserved (when damping = 0).
- At max displacement: all PE, KE = 0
- At equilibrium: all KE, PE = 0
- PE = mgL(1 − cosθ)
- KE = ½mL²ω²
- Total E = mgL(1 − cosθ₀)
With damping, energy is lost per cycle and amplitude decreases exponentially over time.
Finding g from Pendulum
The pendulum is a classic way to determine the local acceleration due to gravity. Rearranging the period formula:
Practical method: Plot T² vs L. The slope of the straight line equals 4π²/g, allowing precise determination of g from the gradient.
This experiment is used in standard physics labs worldwide to verify g ≈ 9.81 m/s² on Earth's surface.
Sources of Error
- Air resistance causes damping and amplitude decay
- Large angles break the small-angle approximation (period increases)
- String elasticity introduces non-ideal behavior
- Human timing error when using a stopwatch
- Bob is not a true point mass (rotational inertia)
- Friction at the pivot increases effective damping
Use small angles (<15°) and average over many oscillations (20–50) for best accuracy.
Planetary Gravity Comparison
- Moon: g = 1.62 m/s² — period ~2.46× longer than Earth
- Mars: g = 3.72 m/s² — period ~1.62× longer than Earth
- Earth: g = 9.81 m/s² — standard reference
- Jupiter: g = 24.79 m/s² — period ~0.63× of Earth
Try switching gravity presets in the simulator and observe how the oscillation speed changes in real time!