Oscillations | Class 11 Physics Notes

13.8 The simple pendulum

A simple pendulum consists of a small bob of mass m suspended from a fixed support by a light, inextensible string of length L. When displaced by a small angle θ from the vertical and released, the bob oscillates about the mean position.

The forces acting on the bob are tension T along the string and gravitational force mg vertically downward. The component mg sin θ acts tangentially to the arc of motion and provides the restoring force causing oscillations.

The torque τ about the support due to the tangential force is τ = -L m g sin θ, where the negative sign indicates the restoring nature of the torque.

Using Newton's law for rotational motion, τ = I α, where I is moment of inertia about the support and α is angular acceleration, we get α = -(m g L / I) sin θ.

For small angles (θ ≤ 20°), sin θ ≈ θ (in radians), simplifying the equation to α = -(m g L / I) θ, which is mathematically identical to the SHM acceleration equation.

For a simple pendulum, I = m L², so angular frequency ω = √(g / L) and period T = 2π √(L / g).

Thus, the motion of a simple pendulum for small angular displacements is simple harmonic with period dependent only on length L and gravitational acceleration g.

An example calculates the length of a pendulum that ticks seconds (period 2 s), yielding L = 1 m for g = 9.8 m/s².

📊 Diagram: Fig. 13.17 (a) A bob oscillating about its mean position; (b) The radial force T - mg cos θ provides centripetal force but no torque; the tangential force mg sin θ provides restoring torque.

🔗 Connection: The simple pendulum's SHM leads to summary and key formulae consolidation in the next section.

Table on page 13 (6×3)

Table on page 5 (5×2)