Question

A simple pendulum is a small mass \(m\) suspended, as shown, by a (weightless) string. Show that for small oscillations (small \(\theta\) ), both \(\theta\) and \(x\) are sinusoidal functions of time, that is, the motion is simple harmonic. Hint: Write the differential equation \(\mathbf{F}=m \mathbf{a}\) for the particle \(m\). Use the approximation \(\sin \theta=\theta\) for small \(\theta\), and show that \(\theta=A \sin \omega t\) is a solution of your equation. What are \(A\) and \(\omega ?\)

Short answer

\theta = A \times sin(\frac{g}{l} t)

Step-by-step solution

Derive the Equation of Motion

For a simple pendulum, the restoring force can be written as \[ F = -mg \times \theta \] when \theta\ is small, where \g\ is the acceleration due to gravity.

Newton’s Second Law

Using Newton's Second Law, \( F = ma \), and substituting \( F = -mg \theta \) and \( a = l \frac{d^2 \theta}{dt^2} \), where \( l \) is the length of the string, we get\[ -mg \theta = m l \frac{d^2 \theta}{dt^2} \].

Simplify the Differential Equation

After canceling \( m \) from both sides of the equation, we obtain\[ l \frac{d^2 \theta}{dt^2} + g \theta = 0 \].

Compare with Simple Harmonic Motion

The simplified equation matches the form of the simple harmonic motion equation, \( \frac{d^2 \theta}{dt^2} + \frac{g}{l} \theta = 0 \). Therefore, we identify \[ \frac{g}{l} = \frac{k}{m} \], where \( \frac{k}{m} = \frac{g}{l} \) is the angular frequency squared \( \frac{g}{l} = \frac{k}{m} \). Solving it, \[ \theta = A \times sin(\frac{g}{l} t) \] gives that the oscillation frequency is \( \frac{g}{l} = \frac{k}{m} \).

Key concepts

Simple Pendulum

A simple pendulum consists of a small mass, usually called a bob, that hangs from a string or rod fixed at one end. This system swings back and forth due to gravity. The bob's movement is a good example of periodic motion.
The strength of the pendulum lies in its simplicity. You only need two things to set it up:

• A weight (the bob)
• A weightless string or rod

When the pendulum is displaced from its equilibrium position and released, it starts to swing. The force of gravity acts on the bob, creating a restoring force that leads it back towards the center.

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes over time. For our simple pendulum, we need to express its motion using these equations.
Starting with Newton's Second Law, which states that force equals mass times acceleration \(F = ma\). For a pendulum, the restoring force derives from gravity and is given by the torque formula for small angles: \[ F = -mg \theta \]
Using this, we substitute \( a = l \frac{d^2 \theta}{dt^2} \) into Newton's Second Law. This results in:
\[ -mg \theta = m l \frac{d^2 \theta}{dt^2} \]
After simplifying, the differential equation of motion for the pendulum becomes:
\[ l \frac{d^2 \theta}{dt^2} + g \theta = 0 \]
This is a standard format for simple harmonic motion.

Newton's Second Law

Newton's Second Law is key to understanding the simple pendulum's motion. It states that the force acting on an object is equal to the mass of the object times its acceleration \(F = ma\). In the case of the pendulum, the force we're concerned with is the restoring force due to gravity.
For small angles, the restoring force is approximated by:
\[ F = -mg \theta \]
This force creates an angular acceleration proportional to the angle of displacement. When we multiply the angular displacement by the mass and gravitational acceleration, the result is the force pushing the bob back towards equilibrium.
By understanding this principle, we derive the necessary differential equations to describe the pendulum's simple harmonic motion.

Small Angle Approximation

The small angle approximation simplifies the mathematics of the pendulum's motion. When \( \theta \) is small (measured in radians), \( \theta \) is approximately equal to \( \text{sin}(\theta) \) and \( \text{tan}(\theta) \). This solves a lot of complexity.
The approximation allows us to write the restoring force simply as:
\[ F = -mg \theta \]
Without this simplification, the exact expression involves more complex trigonometric functions that are harder to solve. Thus, for small displacements, our differential equation for the pendulum becomes:
\[ l \frac{d^2 \theta}{dt^2} + g \theta = 0 \]
This equation is completely analogous to simple harmonic oscillators, showing that the motion is indeed sinusoidal and cyclical.