Class 10 > Unit # 10: General Waves Properties > Simple Pendulum

Simple Pendulum

Video Lecture

Definition

Simple Pendulum: An ideal simple pendulum consists of a point mass (called a bob) suspended from a light, inextensible string tied to a rigid and frictionless support.

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Working of a Simple Pendulum

At the mean position \(O\), the pendulum is in its equilibrium state. When the bob is vertical, the gravitational force \(F_g = mg\) acts vertically downward and the tension \(T\) in the string acts vertically upward. In this equilibrium position, the gravitational force is completely balanced by the tension \(T\).

Figure 1: Resolution of forces acting on a simple pendulum displaced by an angle \(\theta\).

When the bob is displaced slightly through an angle \(\theta\) to an extreme position \(A\) and released, it begins to perform oscillatory motion along a curved arc path of length \(s\).

At position \(A\), two main forces act on the bob:

1. Tension (\(T\)) in the string acting along the length towards the support point.
2. Weight (\(mg\)) acting vertically downward.

The weight \(mg\) can be resolved into two rectangular components:

• \(mg \cos\theta\): Acts along the line of the string, directly opposite to tension \(T\). This component is completely balanced by the tension (\(T = mg \cos\theta\)).
• \(mg \sin\theta\): Acts perpendicular to the string, directed tangentially along the arc back toward the mean position \(O\).

Thus, the component \(mg \sin\theta\) acts as the net restoring force that pulls the bob back towards its mean position.

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Formula for Time Period of Simple Pendulum

The time required by the bob of a simple pendulum to complete one full vibration (or oscillation) is called its Time Period (\(T\)).

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

Where:

• \(T\) = Time Period of the pendulum (seconds, \(\text{s}\))
• \(L\) = Length of the pendulum string from suspension point to the center of the bob (meters, \(\text{m}\))
• \(g\) = Acceleration due to gravity (\(\approx 9.8\text{ m/s}^2\))