Physical Pendulum Calculator
Calculate period, frequency, and oscillation parameters for physical pendulums using moment of inertia
Calculate Physical Pendulum Parameters
Moment of inertia about the pivot point
Total mass of the pendulum
Distance from center of mass to pivot point
Acceleration due to gravity (m/s²)
Physical Pendulum Results
Formula: T = 2π√(I/(mgR))
Where: I = 0.0000 kg·m², m = 0.000 kg, R = 0.000 m
Note: Valid for small oscillations (max angle ≤ 15°)
Comparison with Simple Pendulum
Example Calculation
Uniform Rod Pendulum
Object: Uniform rod of length L = 1.0 m
Mass: m = 2.0 kg
Pivot point: At one end of the rod
Center of mass: At L/2 = 0.5 m from pivot
Moment of inertia: I = (1/3)mL² = (1/3) × 2.0 × 1.0² = 0.667 kg·m²
Calculation Steps
T = 2π√(I/(mgR))
T = 2π√(0.667/(2.0 × 9.81 × 0.5))
T = 2π√(0.667/9.81)
T = 2π√(0.068) = 2π × 0.261
T = 1.64 seconds
Key Formulas
Period
T = 2π√(I/(mgR))
Frequency
f = 1/T
Angular Frequency
ω = 2πf = 2π/T
Radius of Oscillations
L = I/(mR)
Common Objects
Rod (end pivot)
I = (1/3)mL²
R = L/2
Rod (center pivot)
I = (1/12)mL²
R = L/2
Disk (edge pivot)
I = (3/2)mr²
R = r
Sphere (surface pivot)
I = (7/5)mr²
R = r
Physics Tips
Moment of inertia must be about the pivot point
Valid only for small oscillations (≤15°)
Period is independent of oscillation amplitude
Use parallel axis theorem: I = I_cm + md²
Understanding Physical Pendulums
What is a Physical Pendulum?
A physical pendulum is any rigid body that oscillates about a fixed pivot point under the influence of gravity. Unlike a simple pendulum (point mass on a string), a physical pendulum has distributed mass, making its moment of inertia crucial to its motion.
Key Physics Concepts
- •The period depends on moment of inertia, mass, and pivot distance
- •Mass distribution affects the oscillation frequency
- •Energy conservation between kinetic and potential forms
- •Small angle approximation (sin θ ≈ θ) required
Formula Derivation
Starting from torque equation:
τ = Iα = -mgR sin θ
I(d²θ/dt²) = -mgR sin θ
For small angles (sin θ ≈ θ):
d²θ/dt² + (mgR/I)θ = 0
This gives angular frequency:
ω = √(mgR/I)
Radius of Oscillations
The radius of oscillations L = I/(mR) represents the length of an equivalent simple pendulum that would have the same period. This concept helps compare different physical pendulums and understand how mass distribution affects motion.