Physics > Simple Harmonic Motion > 8.0 Force and Energy in Simple Harmonic Motion
Simple Harmonic Motion
1.0 Types of Motion
2.0 Causes of Oscillation
3.0 Solution of the Equation of SHM
4.0 Kinematics of SHM equation i.e., $x = A\sin \left( {\omega t + \phi } \right)$
5.0 Relation between Simple Harmonic Motion and Uniform Circular Motion
6.0 Sign Convention of a Simple Harmonic Motion
7.0 How to Write the Simple Harmonic Motion Equation
8.0 Force and Energy in Simple Harmonic Motion
9.0 Basic Differential Equation of SHM
10.0 Method for Calculating Time Period of a Simple Harmonic Motion
10.1 Restoring Force or Torque Method
10.2 Energy Method
10.3 Important points
10.4 Concept of Pseudo Force
11.0 Spring Block System
12.0 Physical Pendulum
13.0 Vector Method of Combining Two or More Simple Harmonic Motions
14.0 Simple Harmonic Oscillation of a Fluid Column
8.2 Potential Energy ($U$)
10.2 Energy Method
10.3 Important points
10.4 Concept of Pseudo Force
To obtain the potential energy we use the relation. $$\begin{equation} \begin{aligned} F = - \frac{{dU}}{{dx}} \\ \frac{{dU}}{{dx}} = - F \\ \frac{{dU}}{{dx}} = kx\;\left( {as\ F = - kx} \right) \\\end{aligned} \end{equation} $$ Integrating both sides and putting limits we get,
$$\begin{equation} \begin{aligned} \mathop \smallint \limits_0^U dU = \mathop \smallint \limits_0^x kxdx \\ U = \frac{1}{2}k{x^2} \\ U = \frac{1}{2}m{\omega ^2}{x^2} \\\end{aligned} \end{equation} $$
Thus the potential energy has a minimum value at the center ($x=0$), and increases as the particle approaches either extreme of the oscillation ($x=ħA$).
