Module 5: C17 - Oscillations Flashcards
What happens to an object that is displaced from its equilibrium position
It travels towards the equilibrium position at increasing speed. It then slows down once it has gone past the equilibrium position and eventually reaches maximum displacement (amplitude). It then returns towards its equilibrium position, speeding up, and once more slows down to a stop when it reaches maximum negative displacement. This motion is repeated over and over again.
Displacement
-Symbol
-Unit
-Definition
Symbol: x
Unit: m, Meters
Definition:
The distance from the equilibrium position
Amplitude
-Symbol
-Unit
-Definition
Symbol: A
Unit: m, Meters
Definition:
The maximum displacement from the equilibrium position
Period
-Symbol
-Unit
-Definition
Symbol: T
Unit: s, Seconds
Definition:
The time taken to complete one full oscillation
Frequency
-Symbol
-Unit
-Definition
Symbol: f
Unit: Hz, Frequency
Definition:
The number of complete oscillations per unit time.
What is Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM) is a common kind of oscillating motion defined as oscillating motion for which the acceleration of the object is given by:
a = -w^2 x
Equation for acceleration of an object in Simple Harmonic Motion
a = -w^2 x
What are 2 key features of objects moving in Simple Harmonic Motion
- The acceleration a of the object is directly proportional to its displacement x, that is a∝x.
- The minus sign in the equation ‘a = -w^2 x ‘ means that the acceleration of the object acts in the direction opposite to the displacement (it returns the object to the equilibrium position).
Does the period of something in SHM depend on the amplitude (and what happens when amplitude increases)
The period of a simple pendulum moving in SHM does not depend on the amplitude of the swing, so the period does not change. Such an oscillator is referred to as an isochronous oscillator.
What two things must an object be undergoing to be what’s described as Simple Harmonic Motion
- The acceleration of the object is towards a fixed point. In other words the acceleration is in the opposite direction to the displacement.
- The acceleration is proportional to the displacement of the object.
What type of graphs do oscillating objects usually produce
Most examples of oscillation produce graphs in a sinusoidal shape and shows the amplitude A, and period T.
How is the change in velocity shown in a graph (+ when is velocity greatest and lowest on the graph)
The change of velocity over time is given by the gradient of the displacement-time graph
- The velocity is greatest when the gradient of the displacement-time graph is greatest (i.e zero displacement)
- The velocity is zero when the gradient of the displacement-time graph is zero (i.e maximum displacement)
How is the change in acceleration shown in a graph (+ when is velocity greatest and lowest on the graph)
The change of acceleration over time is given by the gradient of the velocity-time graph.
- The acceleration is greatest when the gradient of the velocity-time graph is greatest (i.e zero velocity (greatest displacement in the opposite direction))
- The acceleration is zero when the gradient of the velocity-time graph is zero (i.e Maximum velocity (zero displacement)).
Example Question:
A simple pendulum oscillates in SHM with an amplitude of 32mm. It takes 20s to complete 10 complete oscillations.
Calculate:
a) the Frequency
b) the Initial Acceleration
A = 32x10^-3 = 3.2x10^-2m
t = 20s
n = 10 oscillations
20/10 = 2s per oscillations
a)
f = 1/T => 1/2
f = 0.5Hz
b)
a = -ω^2x
a = -(2π/T)^2 x
a = - (2π/T)^2 x 32x10^-3
a = -0.32ms^-2
What is the displacement x of an object in SHM given by
x = Acosωt (for displacement A at time t=0)
x = Asinωt (for zero displacement at time t=0)
Where A is the maximum displacement
Or
x = Acos(2πft)
x = Asin(2πft)
What is the velocity of an object in SHM given by?
The velocity of an object in SHM is given by the gradient of a displacement time graph. This is the differential dy/dx of x = Asinωt.
The solution is v = ωA cos (ωt)
(Where v is velocity, t is time, ω is angular frequency, A is max. displacement)
What is the maximum velocity of an object in SHM
v = ωA cos (ωt)
Note this will be a maximum when x = 0 and cos(ωt) = 1.
Therefore:
Vmax = ωA = 2πfA
Equation for Displacement, Velocity, and Acceleration of an object in SHM
Displacement = Asin(ωt)
Velocity = ωAcos(ωt)
Acceleration = -ω^2Asin(ωt)
(a = -ω^2x)
How can you calculate the velocity of a simple harmonic oscillator at displacement x? (And also when x=0)
v = ± ω √(A^2 - x^2)
When displacement x = 0
Vmax = ωA
How can you calculate the acceleration of an object in SHM? (And when it’s at maximum acceleration)
a = -(ω)^2 x
When acceleration is at its maximum,
Amax = -(ω)^2 A
(Where ω is the angular frequency, and A is the amplitude (max displacement).)
Example Question:
A spring oscillates in SHM with a period of 3s and an amplitude of 58mm. Calculate:
a) The frequency
b) The maximum acceleration
a)
f = 1/T
T = 3.0s => f = 1/3
f = 0.33Hz
b)
A = 58x10^-3m
f = 0.33Hz
ω = 2πf
ω = 2πf x 0.33
ω = 2.10 rads^-1
a = -(ω)^2 A
a = -(2.10)^2 x 58x10^-3
a = -0.25ms^-2
Example Question:
The displacement of an object oscillating in SHM changes with time and is described by X (mm) = 12 cos 10t where t is the time in seconds after the object’s displacement was at its maximum value.
Determine:
a) The amplitude
b) The time period
c) The displacement at t = 0.10s
a) 12mm
b)
ω = 10 rads^-1
ω = 2π/T => T = 2π/ω
T = 2π/10 => T = 0.63s
c)
12cos(10x0.1) = 6.48mm
Example Question:
An object on a spring oscillating in SHM has a time period of 0.48s and a maximum acceleration of 9.8ms^-2. Calculate:
a) It’s frequency
b) It’s amplitude
a)
T = 0.48s
f = 1/T => f = 1/0.48
f = 2.08Hz
b)
a = -(ω)^2 A
ω = 2πf
ω = 2π x 2.08
ω = 13.9
9.8 = -(13.9)^2 A
A = -0.057m
Example Question:
An object oscillates in SHM with an amplitude of 12mm and a period of 0.27s. Calculate
a) The frequency
b) Its displacement and direction of motion 0.10s, and 0.20s after the displacement was +12mm
a)
T = 0.27s
f = 1/T
f = 1/0.27
f = 3.70Hz
b)
x = Acos(ωt) = Acos(2πft)
x = 12cos(2πx3.7x0.1) = -8.2mm
x = 12cos(2πx3.7x0.2) = - 0.75mm
Definition of Resonance
Resonance is the tendency of a system to oscillate with a larger amplitude at some frequencies than at others.
This particular frequency is called the resonant frequency & at this point the system is said to be in resonance
What is Natural Frequency
The frequency at which a system oscillates without an external periodic force being applied
Periodic Force
A force with a regularly changing amplitude & a definite time period
What are Forced Oscillations
The oscillations of a system which is exposed to an external periodic force
What is Resonance
For a lightly damped system, the amplitude of the oscillations tend to a maximum when the frequency of an applied periodic force is the same as the natural frequency of the system.
If the car suspension consisted only of springs, what would happen to the motion of the car for sometime after hitting a bump in the road?
With only springs the vehicle will continue to bounce (oscillate up and down) for some time after hitting the bump and the passengers would feel sea sick. Think SHM!
How would a pendulum continue to swing in an “ideal physics world” with no friction and a vacuum?
Friction and air resistance reduce the amplitude until eventually the pendulum stops
What will the ride in the car be like when the damping is not strong enough
The oscillation slowly dies away (exponential decay). Oscillations slowly die away
What will the ride in the car be like when the damping is too strong enough
The damping is so strong that the displaced object takes a long time to return to the equilibrium position and does not oscillate
What will the ride in the car be like when the damping is just right
Just enough to stop the oscillation as quickly as possible often about T/4 little or no overshoot
When is an oscillation damped?
An oscillation is damped when an external force that acts on the w oscillator has the effect of reducing the amplitude of its oscillations. For example, a pendulum moving through air experiences air resistance, which damps the oscillation until eventually the pendulum comes to rest.
What would be an example of light damping (+ how does light damping work?)
When the damping forces are small, the amplitude of the oscillator gradually decreases with time, but the period of the oscillations is almost unchanged. This type of damping is referred to as light damping. This would be the case for a pendulum oscillating in air.
What would be an example of heavy damping (+ how does heavy damping work?)
For larger damping force, the amplitude decreases significantly, and the period of the oscillations also increases slightly. This type of heavy damping would occur for a pendulum oscillating in water.
What would be an example of very heavy damping (+ how does very heavy damping work?)
An oscillator such as a pendulum, moving through treacle or oil. In this example of very heavy damping, there would be no oscillatory motion. Instead the oscillator would slowly move towards its equilibrium position.
What is Forced Oscillation (and what is the driving frequency)
A forced oscillation is one in which a periodic driver force is applied to an oscillator. In this case, the object will vibrate at the frequency of the driving force (the driving frequency.)
What is an example of a Forced Oscillation
For example, a mass hanging on a vertical spring can be forced to oscillate up and down at a given frequency if the top of the spring is held and the hand moves up and down. The hand is the driver and it’s motion provides a driver frequency that forces the mass-spring system to oscillate.
What happens when the driving frequency is equal to the natural frequency of an oscillating object
If the driving frequency is equal to the natural frequency of an oscillating object, then the object will resonate. This will cause the amplitude of the oscillations to increase dramatically, and if not damped, the system may break.
What happens when an object resonates?
When an object resonates, the amplitude of the oscillation increases considerably. If the system is not damped, the amplitude will increase to the point at which the object fails.
In the case of the Tacoma Narrows Bridge, the kinetic energy from the wind was efficiently transferred to the bridge, leading to its ultimate collapse.
What 2 things are equal when resonance occurs
For a forced oscillator with negligible damping, at resonance
Driving frequency = natural of the forced oscillator
When does the greatest possible transfer of energy from the driver to the forced oscillator occur at
The greatest possible transfer of energy from the driver to the forced oscillator occurs at the resonant frequency. This is why the amplitude of the forced oscillator is maximum.
What are examples of Resonance in Real Life
- Many clocks keep time using the resonance of a pendulum or of a quartz crystal
- Many musical instruments have bodies that resonate to produce louder notes
- Some types of tuning circuits (for example in car radios) use resonance effects to select the correct frequency radio wave signals.
- Magnetic resonance imaging (MRI) enables diagnostic scans of the inside of our bodies to obtained without surgery or the use of harmful X-rays.
What happened with The Millennium Bridge?
The Millennium Bridge in London was opened in June 2000. It was quickly nicknamed the “wobbly bridge”after it was discovered to resonate when large numbers of people walked across it. As the bridge started to sway, pedestrians tended to match their step to the sway, providing a driving force that was very close to the natural frequency of the bridge. To prevent a possible collapse like that of the Tacoma Narrows, the bridge was closed for two years to allow engineers to install dampers.
Damping a forced oscillation has the effect of reducing the maximum amplitude at resonance. The degree of damping also has an effect on the frequency of the driver when maximum amplitude occurs.
What happens as the amount of damping increases on a graph
- The amplitude of vibration at any frequency decreases
- The maximum amplitude occurs at a lower frequency than f0
- The peak on the graph becomes flatter and broader
Where does the maximum amplitude occur for light damping
For light damping, the maximum amplitude occurs at the natural frequency f0 of the forced oscillator.
