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PH4. Vibrations Simple harmonic motion - s.h.m. a mass bouncing on a spring. Examples include…. …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.
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PH4 Vibrations • Simple harmonic motion - s.h.m.
a mass bouncing on a spring • Examples include….
…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.
Also… • … the acceleration of the body is directly proportional to its displacement from a fixed point • and is always directed towards that point.
Let’s consider a pendulum…(taking positive to be to the right) • Displacement, x = max = amplitude, A • Acceleration, a = max = -amax (left) • Velocity = zero • So… • x = xmax = A • a = -amax • v = 0
As it swings through the centre… • Displacement, x = 0 • Acceleration, a = 0 • Velocity = max = -vmax (left) • So… • x = 0 • a = 0 • v = -vmax
It stops and then… • Displacement, x = max = amplitude, -A • Acceleration, a = max = amax (right) • Velocity = zero • So… • x = xmax = -A • a = amax • v = 0
As it swings through the centre again… • Displacement, x = 0 • Acceleration, a = 0 • Velocity = max = vmax (right) • So… • x = 0 • a = 0 • v = vmax
So the acceleration is always doing what the displacement is doing…they are directly proportional • x = xmax • a = amax • v = 0 • x = xmax • a = -amax • v = 0 • x = 0 • a = 0 • v = vmax • x = 0 • a = 0 • v = -vmax
Similarly with a mass on a spring…
So the defining equation for shm is… • The minus sign means the acceleration and displacement are oppositely directed.
..and the definition in words is… • If 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.
The bigger the amplitude of the oscillation the higher the peak of the sine wave
The usual equations and terms apply T • T (s) = Time Period = time for one oscillation • f (hz) = frequency = no. of oscillations/sec • A = amplitude = maximum displacement from equilibrium position A t
Plotting the pendulum’s displacement displacement time
Consider the pendulum again… • Starting with the pendulum pulled up to the right… • Displacement is a maximum (equal to the amplitude) …and velocity is zero…
Then displacement decreases as… • …velocity increases, but to the left, so in the negative direction.
Putting all three together… • Displacement and velocity we’ve talked about… • …and 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.
Timing your pendulum…(as you do) • You can start timing from when x = A • Or from when x = 0 • These give different displacement-time graphs t=0 x=0 t=0 x=-A
Cosine curve t=0 x=A x = Acos(t + ) t=0 x=0 Sine curve x = Asin(t + )
All the equations!
Period, T (s) = time for one oscillation • Frequency, f (Hz) = number of oscillations per second • Angular frequency, (rad/s) = 2f
The auxiliary circle • Relating circular motion to simple harmonic motion or wave motion
Maximum values VERY important!
Energy -A +A Is she SERIOUS? The Fish diagram! Max k.e. Max p.e. k.e. = 0 Fish eye!! p.e. = 0
a x Graph of acceleration against displacement Think …how are these related? They’re directly proportional to each other … but oppositely directed +2A +A -A -2A
Phase constant, • x = Acos(t + ) is the general solution to the equation d2x/dt2 = - 2x • where 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
Hooke’s Law F = -kx • Force 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 = mg • If it is displaces a distance x below equilibrium the spring tension increases by an amount kx • There is a restoring force kx on the mass F = -kx
Mass on a spring • The motion is simple harmonic because f -x (f will cause a) • a = -2x F = ma so F = -m 2x • Since F = –kx k = m 2 • T = 2/ 2 = k/m so
Damping • When 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.
Damping • The heavier the damping (larger frictional forces) the greater the rate of decay.
Damped oscillations • So 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 oscillations • Free 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.
Resonance • If 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 they’re half way back towards you their oscillation amplitude decreases.
Resonance • A 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 damping • The system, when displaced and released, returns to equilibrium, without overshooting as quickly as possible. • Useful in car shock absorbers – cars mustn’t go into resonant oscillation when they go over bumps in the road. So shock absorbers critically damp the oscillations once they have started.
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