6. Further Mechanics and Thermal Physics 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
- When two objects interact, they exert equal and opposite forces on each other
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 NII?
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
What is centripetal acceleration given by?
a = v2 / r
= ω2 r
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
What is the equation for centripetal force?
F = mv2 / r
= mrω2
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
What is the velocity of the oscillator given by?
v = ±ω √A²-x²
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
What is the equation for maximum velocity?
ωA
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?
- 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
