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PH4VibrationsSimple harmonic motion - s.h.m.

Harmonic Motion

Harmonic motion is periodic (repetitive) motion.

Examples include.

a mass bouncing on a spring

Periodic Motion

Definition of period, TT = time required for one cycle of a periodic motionSI unit: seconds/cycle = sDefinition of frequency, f

SI unit: Definition of angular frequency,

SI unit:

or bungee jumping

and a swinging pendulum

Simple harmonic motion is a special type of repetitive motion..The time period of the oscillation stays the same even if the amplitude varies.The time taken to get from a to b and back to a in all three cases below is the same.

Note with the pendulum that even when it approaches equlibrium it doesnt slow down it simply travels a smaller distance from the point of rest. This is also the case for the tine of the tuning fork. Thus, we can say that any body undergoing simple harmonic motion moves periodically with uniform speed. We can also say that if the tine is moving periodically then the pressure variations it creates will also be periodic.

Maximum displacement at 0 seconds

Maximum displacementafter say, 3 seconds

Maximum displacementafter say, 6 seconds

a

b

a

a

b

b

The time taken to get from position a to b in all three cases is the same

Simple Harmonic Motion

Also the acceleration of the body is directly proportional to its displacement from a fixed pointand is always directed towards that point.

Lets consider a pendulum(taking positive to be to the right)Displacement, x = max = amplitude, AAcceleration, a = max = -amax (left)Velocity = zeroSox = xmax = Aa = -amaxv = 0

As it swings through the centreDisplacement, x = 0Acceleration, a = 0Velocity = max = -vmax (left)Sox = 0a = 0v = -vmax

It stops and thenDisplacement, x = max = amplitude, -AAcceleration, a = max = amax (right)Velocity = zeroSox = xmax = -Aa = amaxv = 0

As it swings through the centre againDisplacement, x = 0Acceleration, a = 0Velocity = max = vmax (right)Sox = 0a = 0v = vmax

So the acceleration is always doing what the displacement is doingthey are directly proportionalx = xmaxa = -amaxv = 0x = xmaxa = amaxv = 0x = 0a = 0v = -vmaxx = 0a = 0v = vmax

Similarly with a mass on a spring

Physics 101: Lecture 21, Pg *

Springs and Simple Harmonic Motion


Springs and Simple Harmonic Motion

So the defining equation for shm isThe minus sign means the acceleration and displacement are oppositely directed.

..and the definition in words isIf the acceleration of a body is directly proportional to its distance from a fixed point and is always directed towards that point, the motion is simple harmonic.

Simple harmonic motion can be characterized by a sine function.

Harmonic Motion (cont)

Harmonic motion can be characterized by a sine function

The bigger the amplitude of the oscillation the higher the peak of the sine wave

Simple Harmonic Motion

We can see that the curves look identical except that one is taller than the otherThese two curves have a different maximum displacement or amplitude

The usual equations and terms applyT (s) = Time Period = time for one oscillationf (hz) = frequency = no. of oscillations/secA = amplitude = maximum displacement from equilibrium positiontTA

Plotting the pendulums displacementdisplacementtime

Simple Harmonic Motion (SHM)

Position versus time for simple harmonic motion is shown as sine or a cosine

Consider the pendulum againStarting with the pendulum pulled up to the rightDisplacement is a maximum (equal to the amplitude) and velocity is zero

The Velocity of SHM

The displacement and velocity functions are plotted here for comparisonNote the maximum magnitude of the velocity function is xm

Then displacement decreases asvelocity increases, but to the left, so in the negative direction.

The Velocity of SHM

The displacement and velocity functions are plotted here for comparisonNote the maximum magnitude of the velocity function is xm

Putting all three togetherDisplacement and velocity weve talked aboutand acceleration and displacement do the same as each other but in opposite directions i.e. both go from max to min but in opposite directions.

So we can see that when the displacement is at a maximum, the acceleration is also at a maximum (but opposite in sign)And when the displacement and acceleration are at a maximum, the velocity is zero

Timing your pendulum(as you do)You can start timing from when x = AOr from when x = 0These give different displacement-time graphst=0x=0t=0x=-A

t=0x=At=0x=0Cosine curveSine curvex = Acos(t + )x = Asin(t + )

Now we see all three functionsThe acceleration function is one-half period (or radians) out of phase with the displacement and the maximum magnitude of the acceleration is 2xm

All the equations!


Simple Harmonic Motion:

x(t) = [A]cos(t)v(t) = -[A]sin(t)a(t) = -[A2]cos(t)

x(t) = [A]sin(t)v(t) = [A]cos(t)a(t) = -[A2]sin(t)

xmax = Avmax = Aamax = A2

Period = T (seconds per cycle)Frequency = f = 1/T (cycles per second)Angular frequency = = 2f = 2/TFor spring: 2 = k/m

OR

At t=0 s, x=A or At t=0 s, x=0 m

Period, T (s) = time for one oscillationFrequency, f (Hz) = number of oscillations per secondAngular frequency, (rad/s) = 2f

Properties of simple harmonic motion

Displacement:

Period T:

Frequency:

Angular frequency:

Units: 1/s = 1 Hz

Velocity:

Acceleration:


Acceleration of particle is proportional to the displacement, but is in the opposite direction (a = - w2x).

Displacement, velocity and acceleration vary sinusoidally.

The frequency and period of the motion are independent of the amplitude. (demo).

The auxiliary circleRelating circular motion to simple harmonic motion or wave motion


What does moving along a circular path have to do with moving back & forth in a straight line (oscillation about equilibrium) ??

R

q

8

7

8

7

x

Maximum values VERY important!


Phase of velocity differs by p/2 or 90 from phase of displacement.Phase of acceleration differs by p or 180 from phase of displacement.

Periodic Motion



SI unit:

The Fish diagram!Max k.e.p.e. = 0Max p.e. k.e. = 0Is she SERIOUS?Fish eye!!

Periodic Motion



SI unit:

Graph of acceleration against displacementThink how are these related?Theyre directly proportional to each other but oppositely directed+A-A+2A-2A

Periodic Motion



SI unit:

Phase constant, x = Acos(t + ) is the general solution to the equation d2x/dt2 = - 2xwhere is a constant phase whose value is determined by the position of the oscillator at t = 0. For example, if x = 0 at t = 0, = -/2 and x = Acos(t - /2) = Asin t

Hookes Law F = -kxForce is proportional to extension.Minus sign because the restoring force and the extension are oppositely directed.K, spring constant, a measure of the stiffness of the spring = F/x i.e. the force required to produce unit extension.

Bouncing spring - is it shm?Suspend a mass m on a spring, pull it down a distance x below equilibrium and release.The weight of the mass is supported by tension in the spring when it is in equilibrium i.e. W = mgIf it is displaces a distance x below equilibrium the spring tension increases by an amount kxThere is a restoring force kx on the mass F = -kx

Mass on a springThe motion is simple harmonic because f -x (f will cause a)a = -2x F = ma so F = -m 2xSince F = kx k = m 2 T = 2/ 2 = k/m so

The block-spring system

The frequency depends only on: - the mass of the block- the force constant of the spring.

The frequency does not depend on the amplitude.

Periodic Motion



SI unit:

Simple harmonic motion/oscillation

Restoring force: F = - kx

Acceleration and restoring force: proportional to x directed toward the equilibrium position

Acceleration:

DampingWhen a bell rings it is transferring energy stored in its oscillation to sound by moving the air around.It does work against frictional forces and is said to be damped.Whenever frictional forces act on an oscillator its total energy will diminish with time so that its amplitude decays to zero.

DampingThe heavier the damping (larger frictional forces) the greater the rate of decay.

Damped oscillationsSo damped oscillations are when the amplitude of the oscillations becomes gradually smaller and smaller as energy is taken out of the system.

Deliberate damping!Damping is deliberately introduced into some systems to prevent continuous oscillations.An example is car shock-absorbers.

Free and Forced oscillationsFree oscillations the system oscillates without any force applied.Its frequency is its natural frequency and there is little or no damping.Forced oscillations the system responds to a regular periodic driving force like continually pushing a child on a swing.

ResonanceIf the frequency of the applied force equals the natural frequency of the system resonance occurs.This is when the system oscillates with MAXIMUM amplitude.If you push the child on the swing each time they reach maximum amplitude their oscillation amplitude increases . If you push when theyre half way back towards you their oscillation amplitude decreases.

ResonanceA good example of resonance is when a singer sings a note of frequency equal to that of the natural frequency of a wine glass.

Resonance and engineering.

Critical dampingThe system, when displaced and released, returns to equilibrium, without overshooting as quickly as possible.Useful in car shock absorbers cars mustnt go into resonant oscillation when they go over bumps in the road. So shock absorbers critically damp the oscillations once they have started.

Damped Oscillations

Critical damping- System no longer oscillates but simply relaxes to the equilibrium position.Overdamping- Damping increased beyond critically damped, system returns to equilibrium position with greater time

Variation of the amplitude of a forced oscillation with driving frequency

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