Simple pendulum Knowledge-Based Study

Study notes

1. A simple pendulum consists of a concentrated mass (called the pendulum bob) suspended by a light, inextensible string of length $L$ from a fixed pivot. When pulled aside by an angle $\theta$ and released, the bob oscillates about its lowest position, known as the equilibrium position.

   Forces Acting on the Bob

   At any angular displacement $\theta$, two primary forces act on the bob of mass $m$:

   • Tension along the string toward the pivot.
   • Gravitational force directed vertically downward with magnitude $mg$.

   Resolving the gravitational force into components along and perpendicular to the string shows that the tension balances the radial component $mg \cos\theta$. The tangential component provides the net restoring force directed back toward equilibrium:

   $F_{\text{restoring}} = -mg \sin\theta$

   The Small-Angle Approximation

   For small angular displacements (typically $\theta \le 10^\circ$ or about $0.17$ radians), the sine of the angle is approximately equal to the angle in radians:

   $\sin\theta \approx \theta$

   Since the arc displacement is $x = L\theta$, we can substitute $\theta = \frac{x}{L}$:

   $F_{\text{restoring}} \approx -\left(\frac{mg}{L}\right)x$

   This equation has the exact form of Hooke's Law, $F = -kx$, confirming that the motion is simple harmonic motion with an effective force constant $k = \frac{mg}{L}$.

   Because the motion approximates simple harmonic motion for small oscillations, the standard relationships of SHM apply directly to the pendulum.

   Angular Frequency

   Using the general formula $\omega = \sqrt{\frac{k}{m}}$ with $k = \frac{mg}{L}$, the angular frequency $\omega$ is:

   $\omega = \sqrt{\frac{g}{L}}$

   Period and Frequency

   The period $T$, which is the time required to complete one full cycle of oscillation, is given by:

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

   The frequency $f$, representing the number of complete oscillations per unit time, is the reciprocal of the period:

   $f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$

   Key Physical Insights

   • Independence of mass: The period does not depend on the mass of the bob. A heavy bob and a light bob with equal string lengths have the same period.
   • Independence of amplitude: For small angles, the period is independent of the initial displacement angle. This property is known as isochronism.
   • Dependence on length: Increasing the length of the string increases the period (for example, quadrupling $L$ doubles $T$).
   • Dependence on gravity: The period is inversely proportional to the square root of local gravitational acceleration $g$. On a celestial body with weaker gravity, the pendulum swings more slowly.

   During oscillation, mechanical energy continuously shifts between gravitational potential energy and kinetic energy in the absence of dissipative forces such as air resistance.

   Energy Breakdown Across Key Positions

   • Maximum displacement (turning points): The bob momentarily comes to rest, meaning kinetic energy is zero. The mechanical energy exists entirely as gravitational potential energy relative to the lowest point: $E = mg \Delta h = mgL(1 - \cos\theta_0)$.
   • Equilibrium position: The potential energy is at its minimum (defined as zero). The bob reaches its maximum speed $v_{\max} = \omega A$, and all mechanical energy is kinetic: $E = \frac{1}{2} m v_{\max}^2$.
   • Intermediate points: The total energy remains constant: $E = \frac{1}{2}mv^2 + mgh$.

   Summary of Conditions for SHM

   A simple pendulum exhibits simple harmonic motion strictly under two conditions:

   1. The amplitude remains sufficiently small so that $\sin\theta \approx \theta$.
   2. Frictional and drag forces are negligible so that mechanical energy is conserved.