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# Simple Harmonic Motion Study Guide

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Updated on Sep 27, 2011

## Introduction

This lesson is an application of the uniform rotational motion on the study of the simple pendulum. We will define period, frequency, and elastic force for the simple pendulum. We will discuss the time dependence of the position, speed, and acceleration. We will also study the mechanical energy of the simple oscillator.

## Simple Harmonic Motion and Elastic Force in a Simple Pendulum

You are already familiar with basic quantities such as frequency, period, and angular speed. We will now see how these quantities fit with a simple harmonic motion.

Simple harmonic motion is the repetitive motion of an object with respect to an origin. If an object is hanging by a spring, it will, under its own weight, come to an equilibrium position and in the meantime extend the spring. The equilibrium position happens due to two forces that are equal and opposite: the gravity on the object and the elastic force (equivalent to the tension in the spring).

W = Felastic

The elastic force is always opposing the change in the size of the spring: So if you pull the spring, the elastic force acts to keep the length small, and if you compress the spring, the elastic force will try to bring the spring back to the initial length.

Mathematically, the expression of the elastic force is summarized as:

Felastic = – k · Δx

where Δx is the extension of the spring due to the hanging mass also known as elongation or displacement (see Figure 16.1).

If the spring in the previous example is further extended and the object is allowed to move, the object will oscillate (symmetrical motion) around the equilibrium position creating a simple pendulum. In the absence of any frictional forces, the object will con tinue to oscillate at the same distance with respect to the equilibrium position, and the same interval will be measured every time the object reaches maximum extension. This spring is considered an ideal spring and the motion simple harmonic motion.

The elastic spring constant is dependent on the material and the strength of the chemical bonds. The unit for the elastic constant can be easily found from the definition of the force:

k = – Felasticx

The SI unit is N/m.

In reality, frictional forces spend the kinetic energy of the object. In addition, the spring itself has a mass that needs to be accelerated, so another amount of energy goes into the spring itself. The harmonic motion will become dampened, and the extension of the string, Δx, will decrease with each oscillation until the pendulum comes to a complete stop.

In this lesson, we will be involved with a spring of negligible mass and frictionless motion.

### Elastic Force–Hooke's Law

The elastic force acting in an ideal spring is proportional to the spring elastic constant k and to the elongation or displacement of the spring from its initial length Δx. The elastic force is opposite to the displacement of the spring (shown by the minus sign). This is also known as Hooke's law.

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