What Is a Simple Pendulum?

A simple pendulum is a classic physics system that demonstrates periodic motion, making it vital for understanding oscillations. It consists of a small, heavy bob suspended by a light, inextensible string from a fixed support, moving in a vertical plane under gravity’s influence.

Concept of a Simple Pendulum

Imagine holding a weight on a string and letting it swing. This back-and-forth movement is an oscillation. The simple pendulum creates a predictable pattern if the displacement angle is small.

The restoring force always acts toward the equilibrium position. It is a result of gravity pulling the bob back towards its resting place, producing simple harmonic motion for small angles.

Forces and Motion in a Simple Pendulum

The bob experiences two forces: tension in the string and gravitational force. The string’s tension counters the radial component of gravity, while the tangential component causes oscillations.

This tangential force equals $-mg \sin \theta$, pulling the bob toward the centre. If the angle $\theta$ is small, $\sin \theta$ approximately equals $\theta$ (in radians), greatly simplifying our physics calculations.

Mathematical Derivation of Pendulum Motion

When the bob moves slightly from its mean position, the equation of motion is $F = -mg \sin \theta$. Displacement along the arc $s$ is related to angle by $s = L\theta$.

For small angles, we approximate:

$F \approx -mg\theta$

Substitute $s = L\theta$ to obtain:

$F \approx -\dfrac{mg}{L}s$

This resembles $F = -kx$, with force constant $k = \dfrac{mg}{L}$, indicating simple harmonic motion.

Equation of Motion and Time Period

The angular equation for a simple pendulum is:

$\dfrac{d^2\theta}{dt^2} + \dfrac{g}{L}\theta = 0$

This matches the simple harmonic oscillator’s equation. The angular frequency $\omega_n$ is then:

$\omega_n = \sqrt{\dfrac{g}{L}}$

The time period $T$ is given by:

$T = 2\pi \sqrt{\dfrac{L}{g}}$

This time period is crucial in many problems. It only depends on string length and gravity, not mass or amplitude (if angle < 15°).

Physical Meaning and Visualisation

Think of a clock’s pendulum or a playground swing. These work on the same oscillation principle as a simple pendulum. The predictable period enables clocks to keep steady time.

As the bob swings, kinetic and potential energy continuously convert into one another, but the total energy stays constant (neglecting air resistance).

Energy Analysis in a Simple Pendulum

At its lowest position, the pendulum has maximum kinetic energy and minimum potential energy, since speed is highest and height is zero.

Kinetic energy at any point is:

$\text{KE} = \dfrac{1}{2} m v^2$

Potential energy is measured from the lowest point:

$\text{PE} = mgh$

Total mechanical energy (neglecting friction) is the sum of kinetic and potential energies at any position during motion.

For deeper understanding, connect this with Work, Energy And Power for related principles.

Derivation of the Simple Pendulum Period Formula

Start with the simplified equation: $\dfrac{d^2\theta}{dt^2} + \dfrac{g}{L}\theta = 0$. Its solution describes simple harmonic behaviour, where the period is:

$T = 2\pi \sqrt{\dfrac{L}{g}}$

This relation shows that the period increases as length increases, but decreases if gravity is stronger.

Linking this with Oscillations And Waves helps in grasping periodic motion concepts.

Key Assumptions and Conditions

A simple pendulum operates under these ideal conditions: the string is inextensible and massless, and air resistance is negligible.

The mass is concentrated in the bob, and oscillations occur in a single vertical plane with a small angular amplitude (usually less than 15°).

Comparison: Simple Pendulum vs Physical Pendulum

A physical pendulum is any rigid body swinging about a horizontal axis, unlike a simple pendulum’s point mass and weightless string setup.

The time period of a physical pendulum uses its moment of inertia. See Torque for rotational dynamics related to physical pendula.

Importance in Experiments and Practical Life

The simple pendulum has been pivotal in determining local gravity variations, especially because its period depends only on $L$ and $g$. This makes it a standard in labs worldwide.

In studies such as the Simple Harmonic Motion laboratory, students measure how length and gravity affect oscillation periods.

• Standard timekeeping in clocks
• Measuring local acceleration due to gravity
• Demonstrating energy conversion principles
• Modeling periodic biological swings (walking)

Numerical Example: Finding the Acceleration Due to Gravity

Suppose a pendulum’s length is $0.750$ m, and its measured period is $1.7357$ s. What is the value of local gravity $g$?

Given: $L = 0.750$ m, $T = 1.7357$ s.

Use the period formula:

$T = 2\pi \sqrt{\dfrac{L}{g}}$

Solving for $g$:

$g = \dfrac{4\pi^2 L}{T^2}$

Substituting the values:

$g = \dfrac{4\pi^2 \times 0.750}{(1.7357)^2}$

Calculating, we get:

$g = 9.83$ m/s$^2$

This result shows how precise measurements of a pendulum can help determine $g$ accurately.

Practice Question for JEE

A simple pendulum has a length of 2 m. What will be its time period on the moon where $g = 1.63$ m/s$^2$?

Understanding the Equation of Motion

The exact equation includes $\sin \theta$. For larger amplitudes, the motion is no longer perfectly simple harmonic, causing the period to increase slightly.

For precise solutions, see advanced topics like "Equation of motion of simple pendulum" in mathematical physics texts.

Energy variations in the pendulum can be connected with topics from Energy In SHM.

Common Mistakes with the Simple Pendulum

One frequent mistake is considering large angles where the approximation $\sin \theta \approx \theta$ fails. Always use small angles ($\theta \lt 15^\circ$) for accuracy.

Another error is ignoring air resistance in very sensitive measurements. Damping can cause gradual loss of mechanical energy over time.

Applications of the Simple Pendulum

• Time period calculation in physics labs
• Calibration of time measuring devices
• Demonstrating resonance and forced oscillations
• Understanding Kinematic Equations in periodic motion

Key Related Physics Topics

- Oscillations
- Simple Harmonic Motion
- Energy in periodic motion
- Measurement of $g$
- Physical and compound pendulums
- Damping and resonance
- Kinematics of rotational systems