Question

A horizontally directed harmonic force \(F(t)=F_{0} \sin \Omega t\) acts on the bob of mass \(m\) of a simple pendulum of length \(L\). Determine the steady state response of the pendulum.

Short answer

Answer: The governing differential equation is \(mL^2\frac{d^2\theta}{dt^2}=-mgL\theta+F_{0}L\sin\Omega t\).

Step-by-step solution

Write down the equation of motion for the pendulum

Begin by writing the equation of motion for the pendulum under the influence of an external harmonic force, F(t) = F0 * sin(Omega * t). Considering Newton's second law, we can write the torque: \(\tau = I\alpha\) where \(\tau\) is the torque, I is the moment of inertia, and \(\alpha\) is the angular acceleration. For the simple pendulum, \(\tau = -mgL\sin\theta + F_{0}L\sin\Omega t\cos\theta\) and \(I = mL^2\). With small oscillations, we can replace \(\sin\theta\) with \(\theta\) and \(\cos\theta\) with 1. Thus, we have the differential equation: \(mL^2\frac{d^2\theta}{dt^2}=-mgL\theta+F_{0}L\sin\Omega t\)

Key concepts

Harmonic Force

Understanding the concept of a harmonic force is crucial when studying systems in motion. A harmonic force is characterized by its sinusoidal nature, meaning it varies with time in a smooth, periodic fashion, much like the oscillations of a swinging pendulum or the vibrations of a guitar string. In the context of the pendulum problem, the harmonic force is defined as \(F(t)=F_{0} \sin(\Omega t)\), where \(F_0\) is the maximum amplitude of the force, and \(\Omega\) is the angular frequency with which the force oscillates.

This kind of force is regularly encountered in physics, particularly in systems that exhibit simple harmonic motion—a type of periodic movement that happens when the restoring force is directly proportional to the displacement and acts in the opposite direction. In the scenario presented, the harmonic force is applied horizontally to the pendulum's bob, inducing an oscillatory response. As students, it is essential to grasp how this force influences the motion of the pendulum in order to predict and understand its steady state response.

Simple Pendulum Dynamics

The simple pendulum, a fundamental concept within physics, refers to a weight (or 'bob') suspended from a pivot, so it can swing freely under the influence of gravity. When discussing simple pendulum dynamics, we consider factors such as the bob's mass (\(m\)), the length of the pendulum (\(L\)), and the gravitational force.

The dynamics become particularly interesting when an external force, such as the harmonic force in our problem, is introduced. A simple pendulum undergoes simple harmonic motion when displaced to a small angle and left to oscillate freely. However, when an additional time-varying external force is applied, the dynamics evolve into a more complex system requiring a modified approach to understanding its motion—in this case, through the equation of motion tailored for the acted-upon pendulum.

Equation of Motion

At the heart of analyzing the pendulum's movement is the equation of motion, which describes how the system evolves over time. The equation of motion for our pendulum, influenced by both gravity and an external harmonic force, is derived using Newton's second law for rotational systems. For small angular displacements, we can linearize the motion (approximate the sine of the angle by the angle itself when measured in radians), leading us to the linear differential equation:

\[mL^2\frac{d^2\theta}{dt^2}=-mgL\theta + F_{0}L\sin(\Omega t)\]
The left side of this equation corresponds to the moment of inertia of the pendulum times the angular acceleration, while the right side comprises two terms: the restoring torque due to gravity and the torque resulting from the external harmonic force. Solving this differential equation provides the theta (\(\theta\)) as a function of time, detailing how the pendulum's angle changes and thus characterizing its steady state response.

Angular Acceleration

Angular acceleration (\(\alpha\)) is a measure of how quickly the angular velocity of an object is changing with time. It is a vector quantity, having both magnitude and direction, and is directly related to the net torque exerted on the object. In the step-by-step solution given, angular acceleration is derived from the equation \(\tau = I\alpha\), where \(\tau\) is the net torque on the pendulum, and \(I\) represents the moment of inertia.

The moment of inertia for a point mass at the end of a rod (as in our simple pendulum) is given by \(I = mL^2\), assuming the mass of the rod itself is negligible. Therefore, by knowing the net torque applied to the pendulum, one can calculate its angular acceleration. This value is vital for predicting how the pendulum will react over time, particularly when subjected to a varying force such as the harmonic force addressed in the problem.