SHM

http://www.animations.physics.unsw.edu.au/ http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmonicmotion.html SHM australia

119 Recall that the condition for simple harmonic motion is:

F = -kxand hence identify situations in which simple harmonic motion will occur

Two conditions are needed for SHM:i)Acceleration (Force = ma) is proportional to displacement (a x)ii)The accelerating force always opposes the displacement (minus sign)

Combining gives:

F = -KxWhere F is the accelerating force (N), K is a constant (N m-1) and x is displacement (m)

The equation describes motion oscillating about a fixed position. Any oscillation where the above equation is true is an SHM oscillator.

A mass on a spring a simple pendulum molecular vibration

Examples of SHM oscillators

http://www2.ess.ucla.edu/~schauble/molecular_vibrations.htm

F = -kx

For a constant mass, mF = ma

Then

ma = -kx

and

a = -k/m x

Both K and m are constantsK is stiffnessM is mass

120 Recognise and use the expressions

a = -ω2x, a = -Aω2 cos ωt, v = Aω sin ωt, x = A cos ωt and T = 1/f = 2π/ω

as applied to a simple harmonic oscillator

The (1D) motion, an object undergoing circular motion (2D) is SHM

http://highered.mcgraw-hill.com/sites/0078458137/student_view0/chapter6/circular_motion_applet.html picture

The horizontal displacement is sinusoidal:X = A cos t

The gradient is velocity (dx/dt):v = - A sin t

The acceleration (dv/dt):a = - A cos t

Notice that since X = A cos t then:

a = - A cos t can be rewritten

a = - 2 X which is SHM

because a x and force (acceleration) opposes displacement

For objects moving in a circle with a constant speed, the time period is found using:

T = 1/f Alternatively since:angular velocity = angle moved through (radians) / time (s)

For an object moving in a circle with an angular speed, , the time period, T, is the time to complete one revolution (2 radians) and is given by:

T = 2π/ω

ω = /t general equation

Then

So

T = 1/f = 2π/ω

121 Obtain a displacement – time graph for an oscillating object and recognise that the gradient at a point gives the velocity at that point

http://www.animations.physics.unsw.edu.au/jw/oscillations.htm#Frequency equations

122 Recall that the total energy of an undamped simple harmonic system remains constant and recognise and use expressions for total energy of an oscillator

http://www.animations.physics.unsw.edu.au/jw/oscillations.htm#Frequency

123 Distinguish between free, damped and forced oscillations

http://www.animations.physics.unsw.edu.au/waves-sound/oscillations/index.html

Free oscillationAn oscillator is said to be free if the total energy of the system is constant. The sum of the potential and kinetic energy is fixed and the maximum displacement remains constant.

Damped oscillationIn a damped system energy is lost from the system. The total energy is reduced, the maximum potential and kinetic energy reduces and the amplitude of oscillation reduces

http://www.animations.physics.unsw.edu.au/jw/oscillations.htm#Frequency damped

Forced oscillationAll systems that oscillate (mass on springs, simple pendulums) have a natural frequency of oscillation. This is the preferred frequency of oscillation.

These systems can be forced to oscillate at other frequencies, but they resist the force when the forced frequency is not equal to the natural frequency.

When the forced frequency matches the natural frequency resonance occurs.

Resonance is an oscillation at the natural frequency which results in a large amplitude oscillation (the system absorbs energy), if the damping is low the amplitudes can become so large as to damage the oscillating system.

http://www.animations.physics.unsw.edu.au/waves-sound/oscillations/downloads/oscillations_Tacoma.html

Tacoma

124 Investigate and recall how the amplitude of a forced oscillation changes at and around the natural frequency of a system and describe, qualitatively, how damping affects resonance

When a system is forced to oscillate over a range of frequencies.The amplitude is the largest around the natural frequency of the system.

If damping is increased the maximum amplitude decreases

If damping is reduced the maximum amplitude increases, the amplitude may become so large that it can break apart the system

125 Explain how damping and the plastic deformation of ductile materials reduce the amplitude of oscillation

Damping removes energy from the system due to the work done on the surroundings as a consequence the amplitude of successive oscillations decreases.

Plastic deformation removes energy from the system due to the work done on the atomic/molecular structure, as a consequence the amplitude of successive oscillations decreases