Simple Harmonic Motion and Hooke’s Law Chapter 13 Simple Harmonic Motion and Hooke’s Law

If the end of the ruler is pulled down through a small Displacement s, the ruler exerts an upward force, called the Restoring Force, trying to pull it back up. If the displacement is not too large the Restoring Force is Directly Proportional to the Displacement and acts in the opposite direction to the displacement.

A spring is hanging vertically A spring is hanging vertically. The length of the spring is called its Natural Length. The spring is stretched beyond its natural length by a Displacement s. The spring then exerts a force trying to restore it to its original length. This force is the Restoring Force F. The Restoring Force is Directly Proportional to the Displacement and acts in the opposite direction to the displacement.

State Hooke’s Law. Hooke’s Law states that when certain elastic objects are stretched or compressed by a displacement s, the Restoring Force is Directly Proportional to the Displacement. OR F   =   k s where: F = the restoring force s = the displacement k = a constant [ k is called the Elastic Constant or the Spring Constant.]

A mass hangs on a spring. There is one position O where the upward force on the mass due to the spring is equal to its weight. If the mass is placed at O at rest it will remain there. O is called the Equilibrium Position. If the mass is pulled down to B and released it vibrates up and down. A particle vibrating up and down like this is said to be moving with Simple Harmonic Motion (S.H.M.).

What is Simple Harmonic Motion? A body moves with Simple Harmonic Motion if: Its Acceleration is Directly Proportional to its Distance from a fixed point on its path and Its Acceleration is always directed Towards that Point.

What is the Equation Defining Simple Harmonic Motion? a is the acceleration s is the displacement ω is a constant

If a system obeys Hooke’s Law then: F   =    k s m a  =    k s ( Since F = m a ) a   =    ​ω​ 2 ​s ( ​ω​ 2 ​ = ​k / m ) the system moves with Simple Harmonic Motion. If a system obeys Hooke’s Law, the system executes Simple Harmonic Motion.

Examples of bodies moving with Simple Harmonic Motion A mass vibrating up and down at the end of a spring can be shown to move with S.H.M. Each prong on a vibrating tuning fork moves with S.H.M. The projection of uniform circular motion on a diameter is S.H.M. For a small angle of swing a pendulum moves with S.H.M. The tides coming in and out every 12 hours move with S.H.M. A magnet suspended horizontally from a piece of thread moves with S.H.M. if it is displaced slightly from being aligned North-South.

Simple Harmonic Motion Q moves around the circle at steady speed. Its shadow moves back and forth with Simple Harmonic Motion

The time taken for the bob to go from A to B and back again to A is the Period of the Pendulum. The Period is the Time for One Complete Oscillation.

If a body moves with S.H.M. whose equation is: then the Period T of the motion is given by:

A body is moving with simple harmonic motion of Period T and Frequency f. What is the relationship between the Period and Frequency?

The Simple Pendulum For a small angle of swing a Simple Pendulum moves with Simple Harmonic Motion.

T is the Period of the Pendulum, i. e T is the Period of the Pendulum, i.e. the time taken for one oscillation. l is the Length of the Pendulum, i.e. the distance from the fixed point of suspension to the centre of gravity of the bob. g is Acceleration due to Gravity.

A Simple Pendulum of Length l has a Period T A Simple Pendulum of Length l has a Period T. Write down a formula for Acceleration due to Gravity g in terms of l and T.

The graph is a straight line passing through the origin. This shows that: l  T 2