Oscillations and Waves

Topic 4.1 Kinematics of simple harmonic motion (SHM)

Examples of oscillations

• A periodic motion is one during which a body continually retraces its path at equal intervals

Nature of oscillating system

p.e. stored as k.e. possessed by moving

Mass on helical spring

Elastic energy of spring

Mass

cantilever Elastic energy of bent rod

Rod

Simple pendulum Gravitational p.e. of bob

Bob

Vertical rod floating in liquid of zero viscosity

Gravitational p.e. of rod or liquid

rod

Amplitude

• Its position x as a function of time t is: where A is the amplitude of motion: the distance from the centre of motion to either extreme

• T is the period of motion: the time for one complete cycle of the motion

• Its position x as a function of time t

OB AP

tAx cos

Amplitude

• Which ball has a larger amplitude?

Ball A

• Which ball has the larger period?

Ball A

Frequency

• The frequency of motion, f, is the rate of repetition of the motion -- the number of cycles per unit time. There is a simple relation between frequency and period:

Tf

1

• If ball B has a time period of 12 s, what is the frequency?

f = 0.0833 Hz

Angular frequency

• Angular frequency is the rotational analogy to frequency. Represented as  , and is the rate of change of an angle when something is moving in a circular orbit. This is the usual frequency (measured in cycles per second), converted

to radians per second. That is

fT

22

• Which ball has the larger angular frequency?

Ball B

• Displayed below is a position-time graph of a piston moving in and out.

Find the:

Amplitude Period

Frequency Angular frequency

10 cm 0.2 s

5.0 Hz 10 rads-1

Phase

• Here is an oscillating ball.

• Its motion can be described as follows:1. Then it moves with v < 0 through the centre to the left 2. Then it is at v = 0 at the left 3. Then it moves with v > 0 through the centre to the right 4. Then it repeats...

Phase

This information concentrates on what phase of the cycle is being executed. It is not concerned with the particulars of amplitude. Mathematically, the phase is the "w t" in:

x(t) = A cos ( t)

Phase

• Here is an oscillating ball.

Recall: x(t) = A cos ( t)  What phase is the ball in when: x = 0, v < 0

A. 0.00 rad B. 0.25 rad C. 0.50 rad D. 1.0 rad E. 1.5 rad F. 1.7 rad

Phase

• Here is an oscillating ball.

Recall: x(t) = A cos ( t)  What phase is the ball in when: x = +A, v = 0

A. 0.00 rad B. 0.25 rad C. 0.50 rad D. 1.0 rad E. 1.5 rad F. 1.7 rad

SHM and circular motion

Uniform Circular Motion (radius A, angular velocity w)

Simple Harmonic Motion (amplitude A, angular frequency w)

• Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis

• The phase angle t in SHM corresponds to the real angle t through which the ball has moved in circular motion.

Velocity and acceleration

• Once you know the position of the oscillator for all times, you can work out the velocity and acceleration functions.

x(t) = A cos (t + )

• The velocity is the time derivative of the position so:

v(t) = -A sin (t + )

• The Change from cos to sin means that the velocity is 90o out of phase with the displacement when – when x = 0 the velocity is a maximum and when x is a minimum v = 0

• The acceleration is the time derivative of the velocity so:

a(t) = -A 2 cos (t + )

• Notice also from the preceding that: a(t) = -2x   • The acceleration is exactly out of phase with the displacement.

Velocity and acceleration

• Watch the oscillating duck. Let's consider velocity now• Remember that velocity is a vector, and so has both negative and

positive values.

• Where does the magnitude of v(t) have a maximum value?

• Where does v(t) = 0?

C

A and E

Velocity and acceleration

• Watch the oscillating duck. Let's consider acceleration now• Remember that acceleration is a vector, and so has both negative and

positive values.

• Where does the magnitude of a(t) have a maximum value?

• Where does a(t) = 0?

A and E

C

Summary 1

• You now know several parameters that are used to describe SHM:

1. amplitude (A)

2. period (T)

3. frequency (f)

4. angular frequency ()

5. initial phase ()

6. maximum velocity (v(t)max) and

7. maximum acceleration (a(t)max)

Summary 2

• There are many relations among these parameters. One minimum set of parameters to completely specify the motion is:

1. amplitude (A)

2. angular frequency ()

3. initial phase ()

• You are already familiar with this set, which is used in:

x(t) = A cos (t + )

• The trick in solving SHM problems is to take the given information, and use it to extract A, and . Once you have A, and , you can calculate anything.

Summary 3

• For a given body with SHM and with the displacement x is given by

txx sin0

txv cos0 xxv 20

txa sin20 xa 2

In terms of time In terms of displacement

Velocity

Acceleration