Question

A simple pendulum consists of a point mass \(m\) suspended by a weightless cord of length \(l .\) Find the equation of motion of the pendulum, that is, the differential equation for \(\theta\) as a function of \(t .\) Show that (for small \(\theta\) ) this is approximately a simple harmonic motion equation, and find \(\theta\) if \(\theta=\theta_{0}, d \theta / d t=0\) when \(t=0\).

Short answer

\( \ddot{\theta} + \frac{g}{l} \sin{\theta} = 0 \rightarrow \) for small \( \theta: \) \( \ddot{\theta} + \frac{g}{l} \theta = 0, \) with solution \( \theta(t) = \theta_0 \cos(\sqrt{\frac{g}{l}} t) \).

Step-by-step solution

Define the problem

A simple pendulum consists of a point mass m suspended by a weightless cord of length l. We need to find the equation of motion for the angular displacement \( \theta \) as a function of time \( t \).

Use Newton's second law

We start by using Newton's second law for rotational motion. The restoring torque \( \tau \) due to gravity is given by \( \tau = -m g l \sin(\theta) \). For rotational systems, \( \tau = I \alpha \) where \( I = m l^2 \) (moment of inertia) and \( \alpha = \ddot{\theta} \) (angular acceleration).

Set up the differential equation

Equating the torque expressions: \( -m g l \sin(\theta) = m l^2 \ddot{\theta} \). Simplifying, we get the differential equation: \( \ddot{\theta} + \frac{g}{l} \sin{\theta} = 0 \).

Approximate for small \(\theta\)

For small angular displacements, \( \sin(\theta) \approx \theta \). The differential equation becomes: \( \ddot{\theta} + \frac{g}{l} \theta = 0 \). This is the equation for simple harmonic motion, \( \ddot{\theta} + \omega^2 \theta = 0 \), where \( \omega = \sqrt{\frac{g}{l}} \).

Solve the simple harmonic motion equation

The general solution of \( \ddot{\theta} + \omega^2 \theta = 0 \) is \( \theta(t) = A \cos(\omega t) + B \sin(\omega t) \). Use the initial conditions \( \theta(0) = \theta_0 \) and \( \frac{d \theta}{dt} (0) = 0 \).

Apply initial conditions

Using \( \theta(0) = \theta_0 \), we get \( A = \theta_0 \). Using \( \frac{d \theta}{dt}(0) = 0 \), we determine \( B \omega = 0 \Rightarrow B = 0 \). Thus, the solution is \( \theta(t) = \theta_0 \cos(\omega t) \).

Key concepts

Newton's Second Law

Newton's Second Law is a fundamental principle that governs the dynamics of objects. It states that the force acting on an object is equal to the mass of the object times its acceleration. In mathematical terms, this is written as: \( F = ma \). For a simple pendulum, we consider the rotational motion, so we need the rotational equivalent of Newton's Second Law. The torque \( \tau \) acting on the pendulum is the product of the gravitational force and the lever arm (length of the pendulum), given by \( \tau = -mgl \sin(\theta) \). Here, the negative sign indicates that the torque acts in the direction opposite to the displacement. According to Newton's Second Law for rotational motion, torque is also equal to the moment of inertia \( I \) times the angular acceleration \( \alpha \): \( \tau = I \alpha \). For a point mass at a distance \( l \), the moment of inertia is \( I = ml^2 \), and the angular acceleration is \( \alpha = \ddot{\theta} \). Combining these, we get: \( -mgl \sin(\theta) = ml^2 \ddot{\theta} \).

Rotational Motion

Rotational motion involves objects that are rotating about an axis. For the simple pendulum, the point mass rotates around the pivot point. The restoring force due to gravity creates a torque that influences this rotational motion. Torque is directly related to angular displacement \( \theta \) and can be expressed as: \( \tau = -mgl \sin(\theta) \). This torque creates angular acceleration, altering the motion of the pendulum. In a rotating system, key quantities include:

• **Angular Displacement (\( \theta \))**: The angle through which the point mass swings.
• **Angular Velocity (\( \omega \))**: The rate at which the angle changes over time.
• **Angular Acceleration (\( \alpha \))**: The rate at which the angular velocity changes over time, given by \( \ddot{\theta} \).

Newton's Second Law can be adapted for rotational motion, connecting torque, moment of inertia, and angular acceleration.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes periodic oscillations where the restoring force is directly proportional to the displacement. For the pendulum, if the angle \( \theta \) is small, the motion approximates SHM. By substituting \( \sin(\theta) \approx \theta \) into the differential equation: \( \ddot{\theta} + \frac{g}{l} \theta = 0 \), we get a standard SHM equation. The general solution for SHM is: \( \theta(t) = A \cos(\omega t) + B \sin(\omega t) \), where \( \omega = \sqrt{\frac{g}{l}} \) is the angular frequency of the oscillation. By applying initial conditions, such as \( \theta(0) = \theta_0 \) (initial angle) and \( \frac{d\theta}{dt}(0) = 0 \) (initial velocity), we find: \( \theta(t) = \theta_0 \cos(\omega t) \). This solution indicates that the pendulum swings back and forth in a harmonic motion with a frequency determined by the length of the pendulum and gravity.

Angular Displacement

Angular displacement is a measure of the change in angle as an object rotates around a pivot point. For the simple pendulum, the angular displacement \( \theta \) is the angle between the vertical and the pendulum's position. It is a crucial variable in describing the motion. The equation of motion derived earlier: \( \ddot{\theta} + \frac{g}{l} \sin(\theta) = 0 \), links angular displacement with other dynamics of the system like acceleration and torque. In SHM, the angular displacement is given by \( \theta(t) = \theta_0 \cos(\omega t) \), showing how \( \theta \) changes over time. Understanding angular displacement helps in predicting the position and behavior of the pendulum at any given moment.

Differential Equations

Differential equations play a crucial role in describing the motion of a simple pendulum. The differential equation we derived: \( \ddot{\theta} + \frac{g}{l} \sin(\theta) = 0 \), encapsulates the relationship between angular displacement, angular velocity, and angular acceleration. To simplify this into a manageable form for small angles, we used the approximation \( \sin(\theta) \approx \theta \), leading to: \( \ddot{\theta} + \frac{g}{l} \theta = 0 \). This is a second-order linear differential equation, characteristic of simple harmonic motion. Solving this gives us the function \( \theta(t) \) describing the pendulum's angular displacement over time. Differential equations are essential for translating physical laws into solvable mathematical forms, which can predict the behavior of oscillatory systems like the pendulum.