Further Mechanics, Thermal Physics & Gases Flashcards
Amplitude of Oscillations
- The amplitude of the oscillations is the maximum displacement of the oscillating object from equilibrium.
- If amplitude is constant and no frictional forces are present then the object is described as freely oscillating/vibrating.
Time period ( Simple harmonic motion)
The time for one complete cycle of oscillation. (starting from a start point and returning back to it)
Angular Velocity
The rate of change of angular displacement with time.
Angular Displacement
For any object in uniform circular motion, the object turns through an angle of 2π/T radians per second
Centripetal acceleration
The acceleration of an object towards the centre of a circle due to a constantly changing velocity at a tangent to the circle.
Centripetal Force
F= mv²/r / F= mω²r
The resultant force of an object in circular motion that always acts towards the centre of the circle.
How do banked tracks prevent the skidding of an object in circular motion?
- For an object travelling in a circular path, there is a centripetal force pushing the object towards the centre of the circle.
- On an a sloped/banked track, the force from the inwards push of the bank (+ any friction) can be provided by the centripetal force
- The support force ( Horizontal component ) will supply the centripetal force rather than the friction so there is no skidding
Why did the object experience a force down on itself when on a big dipper?
On a big dip at high speed you would be pushed in to your seat.
The difference between the support force on the object and it’s weight is the centripetal force
S = mv²/r +mg
Oscillating Motion
Oscillation is the motion back and forth about a fixed point in a straight line.
Define SHM
Simple Harmonic Motion is an oscillating motion caused by a force that repeatedly acts to restore a moving object to its equilibrium position.
Conditions for Simple Harmonic Motion
- The acceleration is proportional to the displacement
- The acceleration is in the opposite direction to the displacement
a = -ω²x
Resonance
When within a forced oscillation system the driving frequency is the same as the natural frequency, causing the amplitude to become large
Resonance is achieved when the phase difference between the driving oscillation and oscillating object is pi/2 ( but at the beginning of the motion the force driver and oscillator’s phase difference is 0)
at a systems natural frequency it is the most stable so the amplitude of the oscillations increase massively
Resonance can only occur when damping is small
Damping Forces
Damping forces are Fricitonal/Dissipative Forces oppose the motion of the oscillating body; they slow or stop simple harmonic motion from occurring.
ALL UNDAMPED OSCILLATORS ARE CLOSED SYSTEMS (energy does not enter or leave)
thus, TOTAL ENERGY = KE +GPE
Light Damping - slowly reduces amplitude of oscillations but keeps the time period almost constant.
Heavy Damping - occurs when the damping is so strong that the displaced object returns to equilibrium much more slowly than if the system is critically damped, no oscillation motion occurs (e.g mass on a spring in thick oil)
Critical Damping - the oscillating system returns to equilibrium in the shortest possible time, with little to no oscillation (e.g mass-spring system vehicle suspension systems)
An object in simple harmonic motion is experiencing light damping due to drag forces, what is the effect of this on the time period?
The object is now 2x in mass how is the frequency of oscillations affected?
T = 2π (L/g) ½
The amplitude of the oscillation does not affect the time period of the oscillation as it is not found in the time period equation.
mass also does not affect the time period because it is independent of the time period for the same reason.
Only things affecting time period are length of pendulum and gravitational field strength.
Equilibrium Position
Position of the lowest energy state
What is Temperature?
Temperature is a measure of the average kinetic energy of the particles in a substance.
- A substance with a high temperature means the particles are vibrating/moving with higher AVERAGE speeds compared with the same substance at lower temperature
What is absolute zero?
When the internal energy of substance is at its minimum (as the kinetic energy u 0)
what is internal energy?
The internal energy of an object is the sum of the random distribution of the kinetic and potential energies of its molecules
First Law of Thermodynamics
The change in internal energy of the object is the total energy transfer due to work done and heating
- due to this when a substance is not heated or cooled it acts as a closed system because no matter or energy is transferred in and out of the system
- this means the substance has a constant internal energy.
- the average speed of the particles in a substance will stay the same provided a closed system
heat transfer
- heat is always transferred from hotter substances to cooler substances
- the particles with more energy transfers some energy to particles with less energy
- heat is transferred by radiation, hotter substance radiate heat quicker than cooler substances
Specific heat capacity
The specific heat capacity is the energy needed to raise 1kg of a substance by 1K without changing state
Specific latent heat
the energy required to change the state of 1kg of a substance without a change in temperature
specific latent heat of fusion - energy required to change 1kg of sold into liquid without coming temperature
specific latent heat of vaporisation - the energy required to change 1kg of a liquid into a gas without changing temperature
Experimental Gas Laws
Boyle’s Law - the inversely proportional relationship between pressure and volume (PV = constant) (isothermal)
P1V1=P2V2
Charles Law - under constant pressure an ideal gas’ volume is proportional to its absolute temperature (PV = T)
work done = pressure x change in volume (on a volume time graph)
V1/T1=V2/T2
any change when pressure stays the same is an isobaric change
any change when pressure is not constant is a
Pressure Law - pressure of a gas of fixed mass and fixed volume is directly proportional to the gas’s absolute temperature (P = kT)
P1/T1=P2/T2
Ideal Gas
and ideal gas is a theoretical gas that obeys the experimental gas laws at all pressures and temperatures
ideal gas assumptions
- all gas particles are identical to each other
- all particle motion in a gas is continuous and random
- all particle motion follow a straight line
- all molecules in a gas have a negligible volume compared to the volume of a container
- the gravitational and electrostatic force between the gas particles does not exist
- the internal energy of a gas has no potential energy store, only a kinetic energy store
- the newtonian laws of motion are obeyed and there are enough molecules to apply statistical laws
- all particle-container collisions are elastic
-
avogadro’s law
V/n = V2/n2
equal volumes of different gases at the same pressure and temperature will contain equal numbers of particles
Boltzman constant
molecular kinetic theory
- the temperature relates to the average speed of the particles within a gas
- ## root mean square speed is different to average speed the average speed of gases in random motion will always be 0
- energy is constantly transferred between the particles within a system, through collisions between the particles
- however, the total combined beefy of the particles is unchanged
- this occurs when a system of particles is isolated/ closed when not heated or cooled
If two materials with different specific heat capacities undergo the same temperature will they have the same internal energy? (assume same potential energy)
- No
- The object with the higher specific heat capacity will store more kinetic energy per temperature change, so will have more internal energy.
- this also means the object will be a larger kinetic energy store and have particles moving, on average at a higher speed
How can we make sure the heat energy transfer from a heat is 100% efficient
- use of insulation material (cladding)
- insuring good thermal contact i.e in the case of water an a heater, using a small amount of oil between them
- the amount of pressure produced is dependent on the rate of change of momentum ( the no. of collision speed second)
- the higher the no. of collisions the greater the pressure
How does Brownian motion prove the existence of atoms?
- Brownian motion is the random motion of particles of varying/range of speeds
-Brownian motion can be observed under a microscope, - it is observed when small particles (such as pollen or smoke particles) suspended in liquid or gas are observed to move around in a random, erratic fashion.
- the collisions between the gas particles and large smoke/pollen particles caused random changes in their speeds and directions
- ( the particles are able to effect their speeds because they travel at much larger speeds and have a lot of momentum)
Derivation of the kinetic theory of gases equation
The derivation is based on:
-the conservation of momentum before an after an elastic collision of a particle ( mv=-mv -> 2mv)
- the pressure of a particle P= F/A
- the average velocity on the three planes of motion x,y,z and how due to the assumptions made by ideal gas law they all equal each other so Crms^2 = 3V(x)^2
= 1/3Nmcrms^2
Outline what is meant by an ideal gas
(for “outline” questions they are looking for combinations of the features of a certain thing)
any from:
- molecules have negligible volume
- collisions are elastic
- there are no interactions between molecules
- the gas obeys the ideal gas laws/obeys Boyle’s law at **ALL pressures and temperatures **
What constitutes Free Vibrations?
- If the amplitude is constant and no frictional forces are present, the oscillations are free vibrations
How to the find the phase difference of two oscillators in motion
phase difference in radians =
2πΔt/T
where Δt is did time between successive instant the two objects are at maximum displacement in the same direction
The relationship between the velocity and the displacement
- the magnitude of the velocity is greatest when the gradient of the displacement-time graph is greatest
- the velocity is zero when the gradient of the displacement-time graph is zero (max displacement)
The relationship between the acceleration and a simple harmonic oscillator
- The acceleration is greatest when the gradient of the velocity-time graph is greatest, (this is when velocity is zero and occurs at maximum displacement in the opposite direction)
- the acceleration is zero when the gradient of the velocity-time is zero. This is when the displacement is zero and the velocity is a maximum
The acceleration is always in the opposite direction to the displacement
Simple Harmonic motion is defined as oscillating motion in which…
1 - acceleration is proportional to the displacement
and
2 - always in the opposite direction to the displacement
a = -ω^2 x
What is the Restoring Force
- The result force on an on
- Adding Mass
- extra mass increases inertia to the system, thus at a given displacement the trolley would be slower. Each cycle of oscillation will therefore take longer
- Using Weaker Springs (low spring constant)
- The restoring force on the trolley at any given displacement would be less, so the trolley’s acceleration and speed at any given displacement would be less. Each cycle of oscillation would therefore take longer (F=ke)
How do you find the acceleration for an object in a mass spring-system?
a = restoring force/mass = -kx/m
For a Free Oscillator, what is the total energy of the system?
As long as friction is absent, the total energy of the system is constant and is equal to its maximum potential energy
Etot = KE + PE
Energy-Displacement Graphs for a Oscillating System
Natural Frequency and Forced Vibrations
- A system that oscillates without a periodic force being applied to it has a frequency = the natural frequency
- when a periodic force is applied to an oscillating system the response depends on the frequency of the periodic force (Forced Vibrations)
What is the effect of increasing the Applied Frequency on a mass-spring system under Forced Vibrations
As the frequency increases,
- the amplitude of oscillations of the system increases until it reaches a maximum amplitude at a particular frequency, and then the amplitude decreases again
- the phase difference between the displacement and the periodic force increases from zero to 1/2 π at the maximum amplitude, the 1/2 π to π as the frequency increases further
Resonance
- when a system is oscillating at the maximum amplitude, the phase difference between the displacement and the periodic force is 1/2 π
- The periodic force is then exactly in phase with the velocity of the oscillating system - resonance
- the frequency at the maximum amplitude is called the resonant frequency
- the lighter the damping
- the larger the maximum amplitude becomes at resonance
- the closer the resonant frequency is to the natural frequency of the system
- the applied frequency becomes increasingly larger than the resonant frequency of the mass-spring system:
- the amplitude of oscillations decreases more and more
- the phase difference between the displacement and the periodic force increases from 1/2 π until the displacement is π radians out of phase with the periodic force
if no damping:
applied frequency of periodic force = natural frequency
at resonance
Which of the pendulums will oscillate with the greatest amplitude when the driver pendulum (D) is oscillating
- The effect of the oscillating motion of D is transmitted along the support thread
- subjecting each of the other pendulums to forced oscillations
- Pendulum R due to being the same length and time period as D, will have a natural frequency = to the natural frequency of D
- R oscillates in resonance with D
- the response of each of the pendulums depend on how close its length is to the length of D,and whether they r shorter or longer than D
Motion Over the top of a hill
- At the top of a hill the support force (S) is directly upwards
- The resultant force on the vehicle is the different between the weight and the support force
mg - S = mv^2/r
the vehicle would lose contact with the road if its speed is equal to or greater than a particular speed
S = 0
What is a Free and Forced Oscillation?
Free - An object oscillates at its natural frequency
Forced - An object is made to oscillate by an external driving frequency
when the driving frequency = the natural frequency, large amplitude oscillations occur (resonance)
How can you prevent resonance from occurring?
- By changing the natural frequency of a system to one that is unlikely to resonate by changing the stiffness of supplies
- introducing damping systems, these do not change the resonant frequency but they do alter the sharpness of the resonant peak
