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Simple Pendulum Calculator

Calculate the period, frequency, and angular velocity of a simple pendulum with interactive visualization

Parameters

mⓘ
°ⓘ
m/s²ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Period:
2.01;s2.01;s
Frequency:
0.50;Hz0.50;Hz
Angular Frequency:
3.13;rad/s3.13;rad/s
Maximum Angular Velocity:
0.82;rad/s0.82;rad/s
Maximum Linear Velocity:
0.82;m/s0.82;m/s

Examples

Example 1: Classroom Pendulum

A pendulum with length 1.0 m and initial angle 15° in Earth's gravity (9.81 m/s²).

  • Period: 2.012.01
  • Frequency: 0.500.50
  • Angular Frequency: 3.133.13

Example 2: Long Pendulum

A longer pendulum (4.0 m) with the same initial conditions.

  • Period: 4.024.02
  • Frequency: 0.250.25
  • Angular Frequency: 1.571.57

Example 3: Moon Pendulum

Same pendulum on the Moon where gravity is 1.62 m/s².

  • Period: 4.934.93
  • Frequency: 0.200.20
  • Angular Frequency: 1.271.27

Visualization

Simple Pendulum

A simple pendulum is one of the most fundamental oscillating systems in physics, consisting of a mass (called the bob) suspended from a fixed point by a string or rod of negligible mass. When displaced from its equilibrium position, the pendulum swings back and forth under the influence of gravity.

The motion of a simple pendulum is approximately simple harmonic for small amplitude oscillations. The restoring force is provided by the component of gravity that acts along the direction of motion, creating a restoring torque that brings the pendulum back to equilibrium.

The period of a simple pendulum is given by T = 2π√(L/g), where L is the length of the pendulum and g is the gravitational acceleration. Remarkably, this period is independent of the mass of the bob and the amplitude of oscillation (for small angles).

The small angle approximation (sin θ ≈ θ) is crucial for the simple harmonic motion behavior. For angles less than about 15°, this approximation is very good. For larger angles, the motion becomes more complex and the period depends on the amplitude.

The simple pendulum has been used historically for timekeeping and is still used in modern physics to measure gravitational acceleration. It demonstrates fundamental principles of oscillatory motion and serves as a model for more complex systems.

Key Concepts

  • Period: Time for one complete oscillation, T = 2π√(L/g)
  • Frequency: Number of oscillations per second, f = 1/T
  • Angular Frequency: Rate of oscillation, ω = √(g/L)
  • Small Angle Approximation: sin θ ≈ θ for θ < 15°
  • Restoring Force: Component of gravity that brings pendulum to equilibrium
  • Simple Harmonic Motion: Oscillatory motion with constant period

Real-World Applications

  • Timekeeping: Historical clocks and metronomes
  • Gravimetry: Measuring gravitational acceleration
  • Seismology: Earthquake detection instruments
  • Education: Demonstrating oscillatory motion principles
  • Engineering: Vibration analysis and damping systems

Explore Further

More mechanics tools

Physics Equations

Period:
T=2πLgT = 2\pi\sqrt{\frac{L}{g}}
Frequency:
f=1T=12πgLf = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}
Angular Frequency:
ω=gL\omega = \sqrt{\frac{g}{L}}
Angular Velocity:
ω(t)=ω0cos⁡(ωt)\omega(t) = \omega_0\cos(\omega t)
Displacement:
θ(t)=θ0cos⁡(ωt)\theta(t) = \theta_0\cos(\omega t)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Angular Frequency

First, we calculate the angular frequency using the formula:

Equation:

ω=gL\omega = \sqrt{\frac{g}{L}}

Calculation:

ω=9.811.00=3.13 rad/s\omega = \sqrt{\frac{9.81}{1.00}} = 3.13 \text{ rad/s}

Explanation:

The angular frequency determines how fast the pendulum oscillates.

2

Step 2: Calculate Period

The period is the time for one complete oscillation:

Equation:

T=2πω=2πLgT = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}

Calculation:

T=2π3.13=2.01 sT = \frac{2\pi}{3.13} = 2.01 \text{ s}

Explanation:

The period is independent of the amplitude (for small angles) and the mass of the bob.

3

Step 3: Calculate Frequency

Frequency is the reciprocal of the period:

Equation:

f=1Tf = \frac{1}{T}

Calculation:

f=12.01=0.50 Hzf = \frac{1}{2.01} = 0.50 \text{ Hz}

Explanation:

Frequency tells us how many oscillations occur per second.

4

Step 4: Calculate Maximum Angular Velocity

The maximum angular velocity occurs when the pendulum passes through the equilibrium position:

Equation:

ωmax=ωθ0\omega_{max} = \omega \theta_0

Calculation:

ωmax=3.13×0.262=0.82 rad/s\omega_{max} = 3.13 \times 0.262 = 0.82 \text{ rad/s}

Explanation:

This is the fastest angular speed the pendulum reaches during its motion.

Frequently Asked Questions (FAQ)

What is a simple pendulum?

A simple pendulum consists of a mass (bob) suspended from a fixed point by a string or rod of negligible mass, swinging under the influence of gravity.

What assumptions are made in simple pendulum calculations?

The calculations assume: small amplitude oscillations (sin θ ≈ θ), negligible air resistance, massless string/rod, and a point mass bob.

How does the period depend on the mass of the bob?

The period of a simple pendulum is independent of the mass of the bob. It only depends on the length and gravitational acceleration.

What happens to the period if the length is doubled?

The period increases by a factor of √2 (approximately 1.414) when the length is doubled, since T ∝ √L.

Why do we use small angle approximation?

For small angles (typically < 15°), sin θ ≈ θ, which makes the motion simple harmonic. For larger angles, the motion becomes more complex and non-harmonic.

Practice MCQs

  1. What is the period of a simple pendulum with length 1.0 m on Earth?
  2. If the length of a pendulum is quadrupled, the period becomes:
  3. The frequency of a pendulum is:
  4. On which planet would a pendulum have the longest period?
  5. For small oscillations, the motion of a simple pendulum is:

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