Section 7 - Further Mechanics Flashcards
What are Newton’s three laws of motion?
- An object stays at rest or in uniform motion unless acted upon by a force
- The rate of change of momentum of an object is proportional to the resultant force acting upon it aka f=ma
- “if object A exerts a force on object B, object B exerts an equal but opposite force on object A”
Symbol for momentum?
p
Unit for momentum?
kg ms-1
How is F=ma derived from Newton’s second law?
- F ∝ Δp / Δt
- F ∝ mΔv/Δt
- F=ma or F=kma
When is F=ma true?
When mass is constant
How would you generally give force using momentum?
F = Δ(mv)/Δt
How would you give NII if mass remains constant?
F = mΔv/Δt or F=ma
How would you give NII if mass is changing?
F = vΔm/Δt
where Δm/Δt is mass change per second
What is the symbol for angular displacement?
θ
What is the unit for angular displacement?
rad
What is the symbol of angular velocity?
ω
What is the unit for angular velocity?
rad s-1
If a wheel takes T seconds to rotate once, what angle will it turn through each second?
2π/T radians
What will the frequency of a rotation of a wheel that takes T seconds to rotate once be?
f=1/T
What is angular displacement given by?
θ = 2πt / T or θ = 2πft
How do we get the equation ω = 2πf?
- circumference = 2πr
- time for one rotation = distance travelled / velocity = 2πr / v or 2π / ω
- so 2πr / v = 2π / ω therefore v = r ω
- also θ = 2πft and ω = θ / t
- ∴ ω = 2πf
The equation for centripetal acceleration is given on the data sheet. what do the symbols stand for?
a = v2 / r = ω^2 r
a = v2 / r = ω^2/ r a= acceleration v= linear velocity r= radius w= centripetal speed
How do we know there must be a centripetal force when a bike wheel is rotating?
It is accelerating as it is changing direction, so there must be a force
Why does kinetic energy stay constant when a bike wheel rotates?
- W=Fd in direction of force
* because F is at 90 degrees to v, no work is done
the equation for centripetal force is found on the data sheet. what do the symbols stand for?
F = mv2 / r = mrω2
F = mv2 / r = mrω2 F= force(N) m=mass(kg) v+ linear velocity r= radius (m)
Is centripetal force a type of force?
No, it is a description of a force - it is a force that causes the acceleration to be towards the centre
When is the centripetal force required increased?
- mass increased
- speed increased
- or radius decreased
For questions about cars going over bumps/hills, what will the fastest speed that the car can travel over the hill be given by?
mg = mvmax2 / r or vmax = √gr
For questions about cars going round bends, what will the maximum friction be given by?
Frictionmax = m vmax2 / r
For questions about cars on banked tracks, what can tanθ be given by?
tanθ = v2 / gr
What are oscillations?
Wobbling about a centre point
What is equilibrium?
The centre point
What is amplitude (A)?
The maximum displacement form centre point
What happens in TSM if there is no friction?
The amplitude is constant (free oscillation)
How does free oscillation occur?
There is no friction, so the amplitude is constant
What is the period (T)?
Time for one complete oscillation
What is the frequency (f)?
The number of oscillations per second (Hz)
How is simple harmonic motion defined?
As oscillating motion where the acceleration is proportional to the displacement in the opposite direction to the displacement
What is acceleration proportional to in SHM?
Directly proportional to displacement from the equilibrium point (a ∝ x)
What are the conditions for SHM?
- acceleration ∝ displacement from the equilibrium point (a∝x)
- displacement and acceleration are in opposite directions (a∝ -x) OR acceleration is always towards equilibrium point
What is displacement (x)?
The distance from the equilibrium position
What is Simple Harmonic Motion a type of?
Oscillation
What is phase difference, ɸ?
The fraction of an oscillation between the position of two oscillating objects
What is phase difference given by?
Δt/T x2π
What is angular frequency, ω?
The rate of change of angular position (given by 2πf)
What is the key equation for SHM?
a = -ω²x
What does it mean, that an oscillator in SHM is an isochronous oscillation?
The period of the oscillation is independent of the amplitude
What does isochronous mean?
Occupying equal time
Graphically, how is the velocity of a pendulum given?
The gradient of a displacement-time graph
Graphically, how is the acceleration of a pendulum given?
By the gradient of a velocity-time graph
In SHM, does T depend on A?
No
When using the equation a=-(2πf)²x, what must be remembered?
2πft must be in radians
How to prove equation for T=2π√m/k?
F = -kx and F = ma
ma = -kx (rearrange for a)
SHM: a = -(2πf)²x
(2πf)² = k/m
1/2π . T = √m/k
T = 2π √m/k
How to prove equation for T=2π√l/g?
(R) Tcosθ = mg
(R) -Tsinθ = ma
Tsinθ / Tcosθ = ma/mg
-tanθ = a/g
small angles tanθ = sinθ (displacement/length = x/l)
-x/l = a/g
(2πf)² = g/l
T/2π = √l/g
T = 2π√l/g
Where is a fiducial marker usually placed in the pendulum experiment?
Equilibrium position
Which two equations can be used to determine the displacement of a simple harmonic oscillator?
- x = A sinωt
* x = A cosωt
What is the difference between waves and oscillations in terms of energy?
Waves transmit energy, oscillations do not
For the equations used to determine the displacement on a simple harmonic oscillator, when is each used?
- x = A sinωt for when oscillator begins at equilibrium position
- x = A cosωt for when oscillator begins at amplitude position
In SHM, where does maximum velocity occur?
At the equilibrium position, with the oscillator being stationary at the amplitude points
In SHM, where does the maximum acceleration occur?
At the amplitude points, and is 0 when the oscillator is at equilibrium position
for the equation what do the symbols stand for?
v = ±ω √A²-x²
v = ±ω √A²-x²
v= velocity
w= angular displacement
A=max displacement
x=displacement
How is the equation for maximum velocity derived from v = ± ω√A²-x²?
Maximum velocity occurs at the equilibrium position, where x=0, so vmax = ωA
the equation for maximum velocity is found on the data sheet. what do the symbols stand for?
vmax=wa
vmax=ωA
vmax= max velocity w= angular frequency A= max displacement
In SHM, what forms of energy are involved?
Kinetic and potential
When does maximum Ek occur in SHM?
At the equilibrium point - where velocity is a maximum
When does maximum Ep occur in SHM?
At the amplitude positions, where displacement is at a maximum
In SHM, is total energy conserved?
Yes
What is damping?
The process by which the amplitude of the oscillations decreases over time
Why might the amplitude of oscillations decrease over time?
Energy loss to resistive force e.g. drag, friction
What happens in SHM if there’s no friction?
Free oscillation
What happens if frictional forces are acting in SHM?
The amplitude decreases
How does the energy-displacement graph for energy in SHM look?
- Ek - ‘sad’ parabola(max. at displacement = 0)
- Ep - ‘happy’ parabolamax at either side where displacement is max.)
- total energy - straight line in line with max. values for Ep and Ek
What are the three types of damping?
- critical
- heavy
- light
What is critical damping?
Object returns to equilibrium in shortest possible time and does not overshoot
What is heavy damping?
Where damping is so strong that object takes a long time to return to equilibrium - oscillation does not happen
Does oscillation occur with heavy damping?
No
How does damping occur in a car suspension system?
- includes spring and damper
* damper provides almost critical damping so that oscillation dies away quickly after going over a bump
How does a displacement-time graph look?
Oscillations start large and then decrease by the same fraction each cycle
What is light damping?
Where T stays the same, and amplitude decreases by the same fraction each cycle (exponentially)
What happens to T in light damping?
Stays the same
What happens to amplitude in light damping?
Decreases by the same fraction each cycle (exponentially)
When does light damping occur?
Naturally (e.g. pendulum oscillating in air)
What happens to amplitude when heavy damping occurs?
Amplitude decreases dramatically
Example of when heavy damping may occur?
Pendulum oscillating in water
Example of when critical damping may occur?
Pendulum oscillating in treacle
What happens to amplitude in critical damping?
The object stops before one oscillation is completed
What is natural frequency?
Frequency at which a SHM oscillator vibrates when no force is applied
What is periodic force?
Regular pushes at the right times to make an SHM oscillator to swing high
When does a system undergo forced oscillation?
When a periodic force is applied
What is forced oscillation?
The oscillation of a system when a periodic force is applied
What happens if applied frequency = natural frequency?
resonance
- amplitude of oscillations becomes very big
- phase difference between displacement and periodic force changes to π/2
- periodic force is now in phase with velocity
What is it called when applied frequency = natural frequency?
The system is resonating
What happens, in terms of applied and natural frequency, when damping is light?
Applied/driving frequency of periodic force = natural frequency of system
What happens, in terms of applied and natural frequency, when damping is heavy?
Resonant frequency is slightly lower than natural frequency
What factors affect resonance?
- when oscillating mass ↑, natural frequency decreases
- when springs are weaker, natural frequency decreases
- damping limits oscillations
What is the link between free oscillation and natural frequency?
When an object is in free oscillation, it vibrates at its natural frequency
What is driving frequency the same as?
Applied frequency
When does resonance occur?
When the driving frequency of the external force is the same as the natural frequency of the object
What will happen when a system is resonating and there is no damping?
The amplitude will continue to increase until the system fails
What will happen when a system is resonating and damping increases?
The amplitude will decrease at all frequencies, and the maximum amplitude occurs at a lower frequency
What is the problem with finding the natural frequency of a system when there is no damping?
It is very hard to do
How can the resonance of an object be investigated experimentally?
- suspend mass between two springs attached to oscillation generator
- ruler placed parallel with mass-spring system to record amplitude
- driver frequency of generator slowly increased from zero - reaching max. amplitude when driver frequency reaches natural frequency of system
- amplitude of oscillation decreases as frequency is increased further
When can a comparatively weak vibration in one object cause a strong vibration in another?
Through resonance
How is g derived from the pendulum experiment?
- T = 2π √L/g - square both sides
- Graph T²-L
- gradient = 4π²/g
- so g = 4π²/gradient
How to verify Hooke’s law from the mass-spring experiment?
- T² = 4π² x m/k
- graph T²-m
- gradient = T²/m = 4π²/k
- draw graph F=kx and see if gradient = 4π²/m
What is 360° in radians?
2π radians
What is 45° in radians?
π/4 radians
What is angular speed?
- The angle an object rotates through per unit time.
- Unit: rad/s
(angle/time)
What is the symbol for angular speed?
ω
What is the symbol for angle?
θ
What is the symbol for time?
t
What is linear speed?
- The speed at which an object is covering distance
* Units: m/s
What is the unit for angular speed?
rad/s
What is the unit for linear speed?
m/s
What is the symbol for radius?
r
What equation gives you the velocity of an object moving in a circular path of a radius
How do you get from that equation to ω = v/r
What are the two basic equations for angular speed?
- ω = θ / t (NOT GIVEN)
* ω = v / r
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant.
why?
All the parts of the particle beam rotate through the same angle in the same time so they have the same angular speed.
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant. The beam of particles in the cyclotrons rotates through 360° in 35μs. Explain why the linear speed of the particles increases as they spiral outwards, even though their angular speed is constant.
Linear speed depends on r, the radius of the circle being turned as well as ω (v = ωr). So, as r increases, so does v, even though ω remains constant.
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant. The beam of particles in the cyclotrons rotates through 360° in 35μs. Calculate the linear speed of a particle at a point 1.5m from the centre of rotation.
- First calculate the linear speed:
- ω = θ / t = 2π / (35 x 10^-6) = 1.7951 x 10^5 rad/s
- Then substitute ω into v = ωr:
- v = ωr = 1.7951 x 10^5 x 1.5 = 2.6927 x 10^5 m/s
- v = 2.7 x 10^5 m/s (to 2 s.f.)
What is angular frequency?
The same as angular speed.
In circular motion, what is frequency?
- The number of compete revolutions per second.
* Units: rev/s or Hz
What are the units for frequency?
Hertz (Hz)
In circular motion, what is period?
- The time taken for a complete revolution.
* Units: s
What is the unit for period?
Seconds (s)
What is the symbol for frequency?
f
What is the symbol for time period?
T
What is the equation linking time period and frequency?
f = 1 / T
What are the three basic equations for angular speed?
- ω = θ / t (NOT GIVEN)
- ω = v / r
- ω = 2πf = 2π / T
Derive the equation that links angular speed and frequency.
• ω = θ / t
• ω = 2π / T = 2πf
or
What is the difference between t and T?
- t is the time
* T is the time period
Are objects travelling in a circle always accelerating?
Yes, because:
• They are changing direction, so the velocity is changing constantly
• Acceleration is defined as rate of change of velocity, so the car is accelerating even if the speed isn’t changing
Which way is an object accelerating when it moves in a circle?
Towards the centre.
What is the force that causes acceleration in a circle and which way does it act?
- Centripetal force
* It acts towards the centre of the circle
What is the acceleration towards the centre of a circle in circular motion called?
Centripetal acceleration
What is the symbol for centripetal acceleration?
a
What is centripetal acceleration?
- The acceleration towards the centre of a circle of an object in circular motion
- Units: m/s²
What are the units for centripetal acceleration?
m/s²
What are the two formulas for centripetal acceleration?
- a = v² / r
* a = ω²r
How are these derived?
a = v² / r
a = ω²r
What is centripetal force (include acceleration)?
- The force that causes the centripetal acceleration and keeps an object moving in circular motion
- Units: N
What is the symbol for centripetal force?
F
What is the unit for centripetal force?
Newtons (N)
Why does centripetal have force (newtons first)?
Newtons 1st = constant velocity unless a force acts on it
Travelling in a circle = changing velocity = acceleration = centripetal acceleration = must be a force causing this
What are the formulas for centripetal force?
- F = mv² / r
* F = mω²r
How are these derived with Newtons second Law?
F = mv² / r
F = mω²r
F=ma a = v² / r a = ω²r F = mv² / r F = mω²r
What would happen if the centripetal force were removed?
The object would fly off at a tangent.
Which way does the centripetal force act?
Towards the centre of the circle.
What does SHM stand for?
Simple harmonic motion
What is the point around which simple harmonic motion occurs called?
Midpoint (equilibrium)
What is displacement in simple harmonic motion?
The distance of an object from the midpoint.
What keeps an object in simple harmonic motion?
A restoring force pulling the object towards the midpoint at all times.
Describe how the restoring force in SHM changes.
It is directly proportional to the displacement.
In SHM, if displacement doubles, what happens to the restoring force?
It also doubles.
Describe the acceleration in SHM.
What is it proportional to and what is it directed towards?
It is directly proportional to the displacement and always directed towards the midpoint.
What is the condition for SHM (acceleration)?
An oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint.
Give the condition for SHM as an equation.
a ∝ -x
Where a = acceleration and x = displacement.
What does the minus sign show?
a ∝ -x
acceleration is always opposing the displacement
What is the symbol for potential energy?
Ep
What is the symbol for kinetic energy?
Ek
What energy exchanges occur in SHM?
Potential energy is transferred to kinetic energy and back.
What type of potential energy (Ep) is involved in SHM?
It depends on what is providing the restoring force.
For a pendulum in SHM, what type of potential energy is involved?
Gravitational
For masses on springs moving horizontally in SHM, what type of potential energy is involved?
Elastic
In SHM, what causes the transfer from Ep to Ek?
The work done by the restoring force.
Describe energy transfers in SHM.
NOTE: The sum of the potential and kinetic energy stays constant.
Describe the energy of an object at maximum displacement in SHM.
- Ep is at its maximum
* Ek is zero
Describe the energy of an object at the midpoint in SHM.
- Ek is at its maximum
* Ep is zero
What is the sum of potential and kinetic energy in SHM called?
Mechanical energy
What is mechanical energy and what happens to it in SHM?
- It is the sum of the potential and kinetic energy.
* It stays constant (as long as the motion isn’t damped).
Describe the graph for Ek and Ep against time in SHM.
- Both curves are the same and are half a cycle out of phase
- Each curve is like a sine wave that has been shifted above the x-axis.
- Note that this means that the wave plateaus slightly each time it reaches the x-axis (i.e. it is not a sharp point at the bottom, but a curve)
Remember to practice drawing out the graph for energy in SHM.
Pg 100 of revision guide
Describe the graph of displacement against time in SHM.
- It is a cosine wave that goes above and below the x-axis.
- Starts at the positive maximum.
- Maximum values are A and -A.
Describe the graph of velocity against time in SHM.
• It is a cosine wave that goes above and below the x-axis.
• Starts at the origin, then goes down.
• Maximum values are ωA and -ωA (where ω = angular frequency and A = amplitude.)
Remember ω = 2πf
(displacement but 90 degrees (π /2) to the left)
Describe the graph of acceleration against time in SHM.
• It is a cosine wave that goes above and below the x-axis.
• Starts at the negative maximum.
• Maximum values are ω²A and -ω²A (where ω = angular frequency and A = amplitude.)
(Reflected displacement)
Why does the acceleration-time graph have minima’s when velocity has maxima’s?
Because it is a stationary point for velocity, meaning it’s derivative (acceleration) is 0
Describe how the graphs for displacement, velocity and acceleration against time in SHM are related (in terms of fractions of wave cycles)
- They are of the same shape, except velocity is 1/4 of a cycle to the left of displacement and acceleration is 1/2 a cycle to the left of displacement.
- Each graph is the gradient of the previous one.
What can be said about the graphs for displacement and acceleration against time in SHM?
The acceleration graph is in antiphase with the displacement. (π/2 radians out of phase)
What are the maximum and minimum values for displacement in a displacement-time graph in SHM?
- Maximum: A
- Minimum: -A
(Where A = amplitude)
What are the maximum and minimum values for velocity in a velocity-time graph in SHM?
- Maximum: ωA
- Minimum: -ωA
(Where ω = angular frequency and A = amplitude)
What are the maximum and minimum values for acceleration in an acceleration-time graph in SHM?
- Maximum: ω²A
- Minimum: -ω²A
(Where ω = angular frequency and A = amplitude)
Remember to practise drawing out the graphs for displacement, velocity and acceleration against time in SHM.
Pg 100 of revision guide
What is a cycle in SHM?
From maximum positive displacement to maximum negative displacement and back.
What is frequency in SHM?
The number of cycles per second.
What is period in SHM?
The time taken for a complete cycle.
What is angular frequency?
- The same as angular speed, except in SHM.
* Equal to 2πf.
In SHM, are frequency and period dependent on amplitude?
No - frequency (and period) are independent of the amplitude
When a pendulum swings, does the amplitude of the swing have any effect on period or frequency?
No - even if it’s swings become very small the pendulum will keep ticking in regular time intervals
What is the defining equation of SHM?
a = -ω²x
How can you convert this into the defining equation of SHM:
a ∝ -x
• Introduce a constant of proportionality
• This depends on ω
• So the equation is:
a = -ω²x
When will it’s speed appear the fastest and slowest?
How do you get the the displacement equation for SHM?
What must be true for the displacement equation to work?
To use this equation you need to start when the pendulum is at it’s maximum displacement - i.e. when t=0, x =±A.
Also you must be in radians
How do you get the acceleration equation for SHM?
How do you get maximum acceleration for SMH
In SHM what would a graph of acceleration against displacement look like?
Add flashcard on how to define angular frequency in SHM.
Do it.
Give the equation for acceleration in SHM.
a = -ω²x
Give the equation for the maximum acceleration in SHM.
a(max) = ω²A
What is the equation for velocity in SHM?
v = ±ωsqrt(A² - x²)
Why is there a plus or minus in this equation:
v = ±ωsqrt(A² - x²)
The object moves in both directions.
If it moves in the positive direction it is +. If it moves in the negative direction it is -.
What is the equation for maximum velocity in SHM?
Max velocity = ωA
What does a graph of Vmax against A look like?
The equation is similar to a circle equation
What is the equation for displacement in SHM?
x = Acos(ωt)
In the equation for displacement in SHM, x = Acos(ωt), at what point is t = 0?
- The object must be at maximum displacement when t = 0.
* i.e. When t = 0, x = A.
What type of motion does a mass on a spring demonstrate?
Simple harmonic motion
When mass is pushed or pulled either side of the equilibrium position on a spring, what is exerted on it?
Restoring force
In a mass-spring system, what is the equation for the force that is used to displace the mass from its equilibrium position?
F = -kx
(Where k = spring constant and x = displacement)
(NOT GIVEN IN EXAM)
How do you get F= -kx?
Give the equation for the period of a mass-spring system.
T = 2πsqrt(m/k)
Where m = mass and k = spring constant
How do you derive T = 2πsqrt(m/k)
What is a mass-spring system?
A mass on a spring that can be displaced from its equilibrium to show SHM.
How do atoms in a solid lattice move?
They vibrate in SHM.
Describe the experiment to check the formula for the period of a mass-spring system.
1) Using string, tie a trolley to a spring
2) Put masses in the trolley
3) Place a position sensor in front of the trolley and spring
4) Pull the trolley to one side by a certain amount and let go.
5) The trolley will oscillate back + forwards as the spring pulls and pushes it in each direction
6) you can measure the period, T, by getting a computer to connect to the position sensor and create a displacement-time graph from a data logger.
7) Read off the period, T, from the graph.
In the experiment investigating a mass-spring system, what types of potential energy are involved in the vertical spring?
Elastic and gravitational. For a horizontal spring, it would be just elastic.
How can you investigate the relationship between mass and Time period of a mass - spring system?
Change the mass (m) by loading the trolley with masses.
Describe the relationship between T and m in a mass-spring system and how this can be shown graphically.
- T ∝ √m
* So a graph of T² against m can be plotted, which shows a straight line of positive gradient.
How could you change the experiment to find the relationship between spring constant and Time period?
Change the spring constant (k) by using different combinations of springs
Describe the relationship between T and k in a mass-spring system and how this can be shown graphically.
- T ∝ √(1/k)
* So a graph of T² against 1/k can be plotted, which shows a straight line of positive gradient.
How can you investigate the relationship between Time period and amplitude of a mass -spring system?
Change the amplitude (A) by pulling the trolley across by different amounts.
Describe the relationship between T and A in a mass-spring system and how this can be shown graphically.
- There is no relationship.
* So a graph of T against A can be plotted, which shows a straight horizontal line.
Describe what graphs can be plotted in an investigation into factors affecting a mass-spring system. What does each illustrate?
- T² against m -> Illustrates T ∝ √m
- T² against 1/k -> Illustrates T ∝ √(1/k)
- T against A -> Shows there is no relationship between T and A.
What factors affect the period of a mass-spring system?
- Mass
* Spring constant
What type of motion does a simple pendulum show?
Simple harmonic motion
What examples of SHM do you need to know about?
- Mass-spring system
* Simple pendulum
Describe how factors affecting SHM in a simple pendulum can be investigated.
1) Attach a pendulum to an angle sensor connected to a computer.
2) Displace the pendulum by a small angle (less than 10°) and let it go.
3) The angle sensor measures how the bob’s displacement from the rest position varies with time.
4) Use the computer to plot a displacement-time graph and read off the period, T, from it. Take the average of several oscillations to reduce percentage uncertainty.
5) Change the mass of the pendulum bob (m), amplitude of displacement (A) and length on the rod (l) independently to see how they affect the period. Plot each graph.
(NOTE: Alternatively, you could also measure the period using a stopwatch. Measure several oscillations and divide by the number to get an average.)
What is a pendulum made of?
- Light, stiff rod
* Pendulum bob
When measuring the period of a simple pendulum, what reference point should be used to measure the start of each oscillation?
The midpoint of the swing, because:
• This is where the pendulum moves fastest, so it is more clear-cut when it reaches the midpoint
• The clamp from which it is hung may be used as a marker
Describe the relationship between T and l in a pendulum and how this can be shown graphically.
- T ∝ √l
* So a graph of T² against l can be plotted, which shows a straight line of positive gradient.
Describe the relationship between T and m in a pendulum and how this can be shown graphically.
- There is no relationship.
* So a graph of T against m can be plotted, which shows a straight horizontal line.
Describe the relationship between T and A in a pendulum and how this can be shown graphically.
- There is no relationship.
* So a graph of T against A can be plotted, which shows a straight horizontal line.
What is the formula for the period of a pendulum?
T = 2π√(l/g)
Where l = length of pendulum (m) and g = gravitational field strength (N/kg)
Derive T = 2π√(l/g)
What are the constraints of the formula for the period of a pendulum?
It only works for small angles of oscillation (up to about 10°).
What is a free vibration?
One where there is (theoretically) no transfer of energy to or from the surroundings.
Remember to revise SHM graphs.
Pgs 102-103 of revision guide.
How does a mass on a spring oscillate when stretched and released?
- At its resonant (natural) frequency.
- It is a free vibration, so if no energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude.
What frequency does the sound from striking metal vibrate at?
The sound you hear is caused by vibrations at the object’s natural frequency.
If you hit a metal object and it vibrates at it’s natural frequency, what happens if it were to have free vibrations?
No energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude forever. In practice this never happens.
Although oscillations can’t happen forever, what do we call a spring vibrating in air anyway?
free vibration
What is a forced vibration?
When a system is forced to vibrate by a periodic external force.
What is the frequency of a driving force causing forced vibrations called?
Driving frequency
What is driving frequency?
The frequency of a driving force causing forced vibrations.
What happens when the driving frequency is much less than the resonant frequency?
The two are in phase since the oscillator just follows the motion of the driver.
What happens when the driving frequency is much greater than the resonant frequency?
The oscillator can’t keep up with the driver, so the two are completely out of phase (antiphase)
What is the resonant frequency of a mass-spring system?
The frequency at which a system oscillates when allowed to vibrate freely.
(CHECK THIS!)
What happens when driving force approaches natural frequency?
System gains more and more energy from the driving force so vibrates with a rapidly increasing amplitude.
The system is resonating.
What is resonance?
- When the driving frequency equals the resonant frequency
- Phase difference between the driver and oscillator is 90°
- Maximum amplitude of vibration is achieved
Describe how amplitude of oscillation changes with driving frequency? Explain with force, energy, phases difference
- When the driving frequency is less than the resonant frequency, the amplitude is low
- As the driving frequency approaches the resonant frequency, the amplitude increases at an increasing rate -> More energy is being transferred from the driving force
- Peak amplitude is at the resonant frequency
- As driving frequency increases above the resonant frequency, the amplitude decreases at a decreasing rate -> The driver and oscillator are completely out of phase, so less energy is transferred from the driving force.
Practice drawing out the set-up to investigate resonance in a mass-spring system.
Pg 104 of revision guide
What is the phase difference between the driver and oscillator at resonance?
90°
Give some examples of resonance.
- Organ pipe - The air resonates when driven by a motion of air in the base. Creating a stationary wave.
- Swing - resonates when it’s driven by someone pushing it at it’s natural frequency.
- Glass smashing - glass resonates when driven by sound wave at the right frequency.
- Radio - tuned so the electric circuit resonates at the same frequency as the radio station you want to listen to.
What is damping?
When an oscillating system loses energy to surroundings.
What causes damping?
Damping forces, such as air resistance.
Give an example of a damping force.
Air resistance
Is damping always unwanted?
No, sometimes systems are deliberately damped to stop them oscillating or to minimise the effect of resonance.
How do shock absorbers in a car suspension work?
They provide a damping force by squashing oil through a hole when compressed.
Is all damping the same?
No, there can be different degrees.
How does damping affect the amplitude of oscillation and how does the degree of damping change this?
The heavier the damping, the quicker the amplitude is reduced.
What is heavy and light damping
Light = long to stop oscillating. The amplitudes reduce by small amounts each period.
Heavy = Less time, amplitudes much smaller per period.
What is critical damping?
• Where the amplitude is reduced to equilibrium in the shortest possible time • Only one sharp displacement is seen
What is overdamping?
- When the damping is too heavy so the system takes longer to return to equilibrium than a critically damped system
- Only one long displacement is seen.
Give an example of overdamping
How can a pendulum on a string and a book on a string represent heavy and light damping?
Give two examples of when critical damping is used.
- Car suspension systems are critically damped so they don’t oscillate but return to equilibrium as soon as possible.
- Moving coil meters
Describe the graphs of displacement against time for light damping and heavy damping.
- Light damping is like a sine wave of gradually decreasing amplitude
- Heavy damping is the same, except the amplitude decreases more sharply
Describe the graph of displacement against time for critical damping.
- Starts at a positive displacement
- The curve is like a gradual and very stretched sine wave until the x-axis, except it begins to flatten out before it reaches the axis
- The line then continues flat along the axis
How do plastic deformation effect amplitude?
reduces amplitude of oscillations the same way as damping.
As the material changes shape, it absorbs energy, so the oscillation will be smaller.
Does damping affect resonance?
Yes
How does damping affect resonance (resonant peak graph)?
- The heavier the damping, the flatter the graph of amplitude against driving frequency will be.
- This means the amplitude at the resonant frequency will be lower.
- The difference in amplitude is most marked near the resonant (natural) frequency.
Describe the set-up used to investigate how damping affects resonance.
- Mass is attached between two springs that are clamped vertically
- Vibration generator is attached to the bottom of the lower spring
- Signal generator is connected to the vibration generator
- Discs of different sizes can be attached to the mass to affect air resistance
- The amplitude of oscillation is looked at by seeing how far the mass moves
Give some used of damping.
- Car suspensions
- Moving coil meters
- Skyscrapers
- Soundproofing
Describe how damping can be used to improve sound performance.
- Sound waves reflect off walls and, at certain frequencies, cause stationary waves to be created between walls of a room.
- This causes resonance and can affect sound quality.
- Soundproofing walls in music studios absorbs the sound waves and converts their energy to heat energy. This prevents the stationary waves from forming.
What is an example of skyscrapers using damping?
What type of damping is this?
