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UNIVERSITY OF SOUTH AUSTRALIA

School of Electrical and Information Engineering (Applied Physics)

INTRODUCTORY PHYSICS, INTRODUCTION TO ENGINEERING PHYSICS & AVIATION PHYSICS 1N

Study Paper 7 Lecture Summary

Vibrations

The simplest type of periodic motion is called simple harmonic motion (SHM).For example, a mass attached to a spring oscillates with SHM.

The force, F s, exerted by a spring on a mass is given by Hooke’s law:

F s = -kx 

where k  is a positive constant called the spring constant. It has units of N/m. The greater the value of k , the greater the

force required to stretch the spring (the “stiffer” the spring).

 x is the displacement of the mass from its equilibrium position ( x = 0).

 Note that F s is directly proportional to  x, but the negative sign indicates that F s and  x are always in opposite directions.

This means that F s is always directed towards the equilibrium position and is known as a restoring force.

SHM occurs when the net force along the direction of motion obeys Hooke’s law.

From Newton’s second law maF   =  

and Hooke’s law, we get makx  =−  

Therefore  xm

k 

a   −=  

SHM is characterised by the following 3 quantities:

The amplitude,  A, is the maximum displacement of the object from its equilibrium position. Unit: m.

The period , T , is the time it takes the object to move through one complete oscillation. Unit: s.

The frequency,  f , is the number of complete oscillations or cycles per unit time. Unit: cycles per second = hertz, Hz.

Frequency and period are inversely relatedT 

 f 1

=  

F s is a conservative force (work done by F s depends only on the starting and finishing points, not on the path between

them).

Therefore a potential energy, known as elastic potential energy, PE s, is associated with F s and is given by2

21 kxPE s   ≡  

PE s represents the energy stored in a stretched or compressed spring or other elastic medium. Unit: J.The total energy stored in the spring is given by2

21 kA E  =  

Because F s is a conservative force, the total mechanical energy,  E , of a mass-spring system is conserved:

 E  = KE  + PE  = constant

Therefore2

212

212

21 kxmvkA   +=

Solving the above for v 

( )22 x A

m

k v   −±=  

The above shows that v = 0 at the extreme positions when  x =   A± and that v is a maximum at x = 0, given by

 Am

k v   ±=

max 

When an object is moving with SHM, its position as a function of time is given by

( ) ft  A x   π  2cos=  

or ( )t  A x   ω cos=  

where ω  (“omega”) is known as the angular frequency and has units of rad/s.

From the above two equations ω  = 2π   f  

The velocity and acceleration as a function of time are given by

( ) ft  Av   π  ω  2sin−=  

( ) ft  Aa   π  ω  2cos2

−=  

The  period, T , of an object of mass m moving with SHM while attached to a spring of spring constant k  is

k 

m

T    π  2=  

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Since frequency,  f , and period, T , are inversely related,  f  is

m

k  f 

π  2

1=  

 Note that in SHM, the period, T, and frequency, f, do not depend on the amplitude, A. 

A simple pendulum consists of a small mass, m, suspended by a light string of length  L fixed at its upper end.

If the mass is displaced by a small angle from the vertical and released, it undergoes SHM.

The restoring force is the force of gravity.

The period of a simple pendulum is given by

g

 LT    π  2=  

and the frequency by

 L

g f 

π  2

1=  

 Note that frequency,  f, and period, T, of a simple pendulum do not depend on mass, m,  but do depend on the

 acceleration due to gravity, g.

In all real mechanical systems, forces of friction retard the motion, reducing the amplitude with time until the

oscillations stop.

Such motion is known as damped oscillation.