Join our WikiEducator discussion group or Register now for free skills training.

From WikiEducator

(: Hi Richard, whouldn't you prefer to use Math and Special Characters?. Please see Help:Displaying_Special_Characters :-). --chela5808 17:06, 2 January 2009 (UTC))

OBJECTIVES

Objectives

By the end of this topic the reader should be able to; define simple harmonic motion (s.h.m), show that a mass oscillating on a spring system executes s.h.m, derive an expression for the period of motion in each example of s.h.m, solve the equation \frac{d^2y}{dt^2}\ subject to the given initial conditions, apply the conservation of mechanical energy to s.h.m.

DEFINITION

Definition

Simple harmonic motion is that kind of periodic motion in which the acceleration of the body along the path of motion is directed towards a fixed point in the line of motion and directly proportional to the displacement of the body from a fixed point.

EXAMPLES OF SIMPLE HARMONIC MOTION

Mass spring systems

Mass attached to spring on a horizontal smooth surface

Mass between two springs on a horizontal smooth surface

Mass attached to a vertical spring

Mass attached to two vertical springs connected in parallel

Mass attached to two vertical springs connected in series

Simple pendulum

Liquid oscillating in a U-tube

Applications of s.h.m

Revision questions

Basic definitions

Activity

Consider a mass attached to an inelastic string rotating in a horizontal circle of radius r and having its centre at O.

Suppose the mass is initially at A and that after a time t it is at B. During this time the radius r of the circle sweeps through an angle θ. The motion of the mass from A round the circle is equivalent to the motion of a mass from A to C and then back to A. The distance X is the displacement of the mass from the centre of motion at any time t. The distance OB ( r ) is theamplitude of motion.

Definition

Amplitude is the maximum displacement of the mass from the centre of motion.

Activity

Consider a mass attached to an inelastic string rotating in a horizontal circle of radius r and having its centre at O.

The acceleration in s.h.m is expressed as  =-ω2x,……………………………………………………..(i)
where ω is the angular velocity.

Definition

’’Angular velocity is the rate of change of angle with time’’

Solving the differential equation (i) yields two solutions; X = rsin( ωt +Є )…………………………….................................(ii)

and X =rcos( ωt+Є )……………………………………………………………………………………………………………..(iii)

which clearly are functions of t. where Є is a constant called the phase constant whose values are determined by the initial conditions. If we consider X= rcos(ωt+Є) and assume that

• If X = r at t=0, then cosЄ = 1 this implies that Є=0, in this case (iii) simplifies to X=rcos ωt.
• If X =0 at t = 0, then cosЄ = 0this implies that Є =90degrees or .

(iii) then becomes X = rcos( ωt + 90 )=- rsinωt. Thus the motions of X=rcosωt and X = rsinωt differ in phase by 90degrees. A sketch of X= rcosωt against time t is of the form: