Y2: Option C - Engineering Physics Flashcards
What is Inertia
A measure of how much an object resists a change in velocity
What is the moment of inertia
A measure of how difficult it is to rotate, or change the rotational speed of an object
What is the equation for the moment of inertia for a point mass
I = mr^2
I: Moment of inertia (kgm^2)
m: Mass (kg)
r: Distance from the axis of rotation (m)
What is the equation for the moment of inertia for an extended object
The moment of inertia is calculated as the sum of all the individual moments of inertia, of each point mass that makes up the object.
∴ I = Σmr^2
How does the distribution of an object’s mass alter it’s moment of inertia
For a point mass, I = mr^2
∴ I ∝ r^2
∴ The moment of inertia for a point mass is greater, if it is further form the axis of rotation
A spinning object can be modelled as a collection of point masses
∴ If the same mass is distributed further from the axis, the overall moment of inertia will increase
What is the equation for the moment of inertia for a hollow ring (hoop)
I = mr^2
What is the equation for the moment of inertia for a solid wheel
I = 1/2(mr^2)
What is the equation for the moment of inertia for a hollow sphere
I = 2/3(mr^2)
What is the equation for the moment of inertia for a solid sphere
I = 2/5(mr^2)
What is the equation for rotational kinetic energy
Ek = 1/2(Iω^2)
For linear motion, Ek = 1/2(mV^2)
For a rotating object, ω = V/r
∴ V = ωr
∴ Ek = 1/2(m(ωr)^2)
∴ Ek = 1/2(mr^2(ω^2))
I = mr^2
∴ Ek = 1/2(Iω^2)
What is angular displacement (θ rad)
The angle through which a point has been rotated
What is angular velocity (ω rads^-1)
The vector quantity describing the angle an object rotates through each second
∴ ω = Δθ/Δt
What is angular speed (ω)
The scalar magnitude of the angular velocity
What is the angular acceleration (α rads^-2)
The rate of change of the angular velocity
∴ α = Δω/Δt
What is the equation that relates linear (a) and angular (α) acceleration
a = αr
α = Δω/Δt
ω = V/r
∴ α = (1/r)(ΔV/Δt)
a = ΔV/Δt
∴ α = (1/r)a
∴ a = αr
What are the quantities for rotational motion that correspond with linear motion
S ⇒ θ
U ⇒ ω1
V ⇒ ω2
A ⇒ α
T = T
What are the equations for rotational motion (SUVAT equivalent)
ω2 = ω1 + αt
θ = 1/2(ω2+ω1)t
θ = (ω1)t - 1/2(αt^2)
θ = (ω2)t + 1/2(αt^2)
(ω1)^2 = (ω2)^2 + 2αθ
What is given by the gradient of an ‘Angular displacement-time’ graph
Gradient = ω
What is shown by a straight line on an ‘Angular displacement-time’
ω is constant (α = 0)
What is shown by a convex ‘Angular displacement-time’
+α (acceleration)
What is shown by a concave ‘Angular displacement-time’
-α (deceleration)
What is given by the gradient of an ‘Angular velocity-time’ graph
Gradient = α
∴ + grad = acceleration
∴ - grad = deceleration
What is given by the area under the curve of an ‘Angular velocity-time’ graph
Area = θ
What is shown by a convex ‘Angular velocity-time’
Increasing α
What is shown by a concave ‘Angular velocity-time’
Decreasing α
What is a couple
A pair of forces that cause no resultant linear motion, but which cause an object to turn
What is torque
The turning effect of a force (or couple) on an object
What is the equation relating torque and perpendicular force
T = Fr
F: force
r: perpendicular distance to the axis of rotation
∴ T = Fr cosθ
What is the equation relating torque, and the moment of inertia of the rotating object
T = Iα
T = Fr, and F=ma
∴ T = mar
a = αr
∴ T = α(mr^2)
I = mr^2
∴ T = Iα
What is the equation for the work done to rotate an object
W = Tθ
For a linear system, W = Fs
For a rotation, s = θr
∴ W = Frθ
T = Fr
∴ W = Tθ
What is the equation for rotational power
P = Tω
Power = rate of change of energy (Work done)
∴ P = ΔW/Δt
W = Tθ
∴ P = Δ(Tθ)/Δt
ω = Δθ/Δt
∴ P = Tω
What is frictional torque
The opposing torque experienced by a rotating system due to the friction of it’s components
T(net) = T(applied) - T(friction)
What is a flywheel
A heavy wheel with a high moment of inertia, to resist changes in rotational motion
How is energy stored in a flywheel
Flywheels are charged as they as spun, with the input torque stored as rotational kinetic energy.
(just enough power is constantly supplied to keep the flywheel fully changed until the energy is required)
What factors effect the energy storage capacity of a flywheel
- Mass:
(Ek ∝ I), and (I ∝ m)
∴ Ek ∝ m - Mass distribution:
(Ek ∝ I), and (I ∝ r^2)
∴ Ek ∝ r^2 - Angular speed:
Ek ∝ ω^2
What are the practical limits of increasing the energy storage capacity of a flywheel
- May be impractical
eg. large, heavy wheel may not fit into the machine - May damage the machine
eg. Increasing ω will increase the centrifugal force, potentially damaging the mechanism
How is the energy storage capacity of modern flywheels improved
- Usually carbon fibre
(although they have a lower mass, can reach much higher speeds without causing damage) - Friction is reduced
(eg. lubrication, levitating wheel with superconductors, work in a vacuum, etc)
How may a flywheel be used in a system
- Smoothing torque:
If input power varies, flywheels uses spurts of energy to change up, and supply constant torque to the system as they decelerate - Smoothing angular velocity:
A charged flywheel will maintain the angular velocity of rotating components when the power supply stops - Supplying additional energy:
If the system is required to exert varying forces, a constant power input can charge a flywheel which will decelerate to release spurts of energy
What are some examples of flywheel applications:
- Potter’s wheel
- Regenerative braking
- Power grids
- Wind turbines
- Riveting machine
What are flywheel batteries
Machines designed to store as much energy as possible
(as much Ek as the ω, r and m of the wheel allows)
What are the advantages of flywheels
- They are very efficient
- They last a long time without degrading
- Recharge time is short
- React and discharge quickly
- Environmentally friendly
What are the disadvantages of fly wheels
- Much larger and heavier than other storage methods
- Pose a safety risk as the wheel could break apart at high speeds (protective casing increases weight)
- Energy lost through friction
- Can oppose direction changes in moving objects
What is the angular momentum of a rotating object
The product of the moment of inertia and angular velocity of a rotating object
∴ Angular momentum = Iω
(Angular version of mv)
What is the law of the conservation of angular momentum
When no external force is applied (torque, friction, etc), the total angular momentum of a system remains constant
∴ I1ω1 = I2ω2
What is angular impulse
The change in angular momentum
= Δ(Iω)
What is the equation that links torque and angular impulse
TΔt = Δ(Iω)
T = Iα
α = = Δω/Δt
∴ T = Δ(Iω)/Δt
(⇒ ∴ T = rate of change of angular momentum)
∴ TΔt = Δ(Iω)
What is translational motion
The movement of the centre of mass of an object
What is rotational motion
The rotation of an object about it’s centre of mass
What is rolling
A combination of rotational and translational motion
What is the equation relating GPE and Ek for a rolling object
Ep = Ek(trans) + Ek(rot)
∴ Ep = 1/2(mV^2) + 1/2(Iω^2)
What is the equation for the linear velocity of a rolling object
V = √((2Ep)/(m+(I/r^2)))
Ep = Ek(trans) + Ek(rot)
∴ Ep = 1/2(mV^2) + 1/2(Iω^2)
ω = V/r
∴ Ep = 1/2(mV^2) + 1/2(I(V/r)^2)
∴ V = √((2Ep)/(m+(I/r^2)))
What is a system in thermodynamics
A volume of space filled with a gas
What is the difference between an open and closed system
- Open system: Gas can flow in/out
- Closed system: Gas can’t enter or escape
What is the first law of theromdynamics
Heat supplied to a system either increases the internal energy of the system or enables it to do work
∴ Q = ΔU + W
Q: Energy transferred to the system by heating
ΔU: Change in internal energy
W: The work done by the system
What does the +/- sign indicate about Q, for the first law of thermodynamics.
- If energy is transferred to the system (system is heated), Q is POSITIVE
- If energy is transferred away from the system (system is cooled), Q is NEGATIVE
What does the +/- sign indicate about ΔU, for the first law of thermodynamics.
- If the internal energy of the system increases, ΔU is POSITIVE
- If the internal energy of the system decreases, ΔU is NEGATIVE
What is a non-flow process
Changes to a gas in a closed system, as air can’t flow in or out
What is the relationship between temperature, pressure and volume, during all non-flow processes for an ideal gas
(p1V1)/(T1) = (p2V2)/(T2)
For an ideal gas, pV = nRT
R is constant (Molar gas constant = 8.31 JK^-1mol^-1)
n is fixed as change occurs in closed system
∴ pV/T = Constant
∴ p1V1/T1 = p2V2/T2
What is an isothermal change
A change that occurs at a constant temperature
(Assumed for slow changes)
What is the equation for the first law of thermodynamics, during an isothermal change to an ideal gas
Q = W
For an ideal gas, U ∝ T
T = constant
∴ ΔU = 0
∴ Q = 0 + W
∴ Q = W
If energy is applied to a system, it will result in an equivalent amount of work done (and vice versa)
What is the relationship between pressure and volume, during an isothermal change for an ideal gas
p1V1 = p2V2
(Boyle’s Law)
What is an adiabatic change
A change that occurs with no heat transfer in our out of the system
(Usually assumed for Fast changes, as there isn’t enough time for heat to be transferred)
What is the equation for the first law of thermodynamics, during an adiabatic change to an ideal gas
ΔU = -W
No heat is transferred
∴ Q = 0
∴ 0 = ΔU + W
∴ ΔU = -W
Any work done will result in an opposite change to the internal energy (and vice versa)
What is the relationship between pressure and volume, during an isothermal change for an ideal gas
p1(V1)^𝛾 = p2(V2)^𝛾
𝛾: adiabatic constant
What is the equation for work done during changes at a constant pressue
W = pΔV
Work done = Force x distance ⇒ W=FΔx
Pressure = Force / Area ⇒ F=pA
∴ W = pAΔx
AΔx = ΔV
∴ W = pΔV
What is the relationship between temperature and volume, during a change at a constant pressure for an ideal gas
V1/T1 = V2/T2
(Charles’ Law)
What is the equation for the first law of thermodynamics, during a change at a constant volume
Q = ΔU
If volume is constant, no work is done (W=0)
∴ Q = ΔU + 0
∴ Q = ΔU
Any heat transferred to or from the system directly changes the internal energy of the gas
What is the relationship between temperature and pressure, during a change at a constant volume for an ideal gas
P1/T1 = P2/T2
(Pressure Law)
What is a pV diagram and what does it show for a non-flow process
- Graph showing the relationship between pressure and volume during the change
- Arrow can be drawn on the curve to indicate the direction of the change
What does the area under a pV graph show for a non-flow process
Work done
What is an isotherm and what does it show
Isotherm = Curve on pV diagram for isothermal change
- Shows p ∝ 1/V, as PV=constant
- T is constant, but curve approaches closer to origin if T is lower (as work done is less)
What is the difference between a pV graph for an isothermal and adiabatic change
The curve for an adiabatic change has a steeper gradient than for an isothermal change
What is the difference between the work done for an isothermal and adiabatic change
- More work is required for adiabatic compression than for isothermal compression
- More work is required for isothermal compression than for adiabatic compression
(larger area under pV curve = More work done)
What is the difference between the gradient of a pV graph for an isothermal and adiabatic process
Isothermal:
pV = constant
∴ let pV = k
∴ p = kV^-1
∴ dp/dV = -kV^-2
k = pV
∴ dp/dV = -(pV)V^-2
∴ dp/dV = -p/V
Adiabatic
pV^𝛾 = constant
∴ let pV^𝛾 = k
∴ p = kV^-𝛾
∴ dp/dV = -𝛾kV^-𝛾-1
k = pV^𝛾
∴ dp/dV = -𝛾(pV^𝛾)V^-(𝛾+1)
∴ dp/dV = -𝛾p/V
Therefore, the gradient of an adiabatic curve is steeper than an isothermal curve, by a factor of 𝛾
What does a pV graph look like for a change at a constant volume
Vertical line
∴ W = 0 (no area under graph)
What does a pV graph look like for a change at a constant volume
Horizontal line
(Work done as volume changes)
What is a cyclic process and how is it shown on a pV graph
If a system undergoes different processes one after another, and returns to the original temperature and pressure before begining the processes again.
This is shown by a closed loop on a pV graph
What is the net work done during a cyclic process and how can it be determined from a pV graph
The difference between the work done by the system and the work done on the system
∴ Shown by the area inside the loop on a pV graph
What is a 4 stroke engine
An internal combustion engine where fuel is burned once every 4 strokes (1 stroke = up or down)
What is the 4-stroke cycle of a petrol engine
1) Induction
- Piston starts at the top and moved down, drawing in an air-fuel mix through the inlet valve
- Constant pressure (just below atmospheric)
2) Compression
- Inlet value closes and piston moves up, compressing the air (Increased p, decreased V)
- Just before the end of the stroke, the spark plug ignites the fuel causing a sudden temp and pressure increase, at almost constant volume
3) Expansion
- The hot gas expands, and does work on the piston pushing it down (decreased p, increased V)
- Work done by gas to expand is greater than work done on gas to compress it, as temp is higher
(∴ Net output work drives engine)
- Just before the end of the stroke, the exhaust valve opens and the pressure decreases
4) Exhaust
- The piston moves up and the burned gas leaves through the exhaust valve
- Constant pressure (just above atmospheric)
How is the 4-stroke cycle of a diesel engine different to a petrol engine
Undergo the same 4-stroke cycle with several slight differences for diesel:
- Only air drawn in during induction
- No spark plug: Air compressed until temp high enough for ignition, then diesel sprayed in by fuel injectors
- No sharp peak before expansion, as injected fuel heats up, increasing the volume at an almost constant pressure, before expansion occurs
What is an indicator diagram
A pV graph showing the changes experienced by the system during a 4-stroke engine cycle
What is a theoretical indicator diagram
pV graph for the cycle of a 4-stroke engine, if it were to occur under perfect conditions
What is an Otto cycle
The theoretical cycle for a 4-stroke petrol engine
What is a Diesel cycle
The theoretical cycle for a 4-stroke diesel engine
What assumptions are made about the system for a theoretical indicator diagram
- The same gas is taken continuously around the cycle
- Pressure and temperature changes are instantaneous
- Heat source is external
- Engine is frictionless
What are the 4 stages of an Otto cycle
1) Gas is compressed adiabatically
2) Heat is supplied whilst volume is constant (increase p)
3) Gas is allowed to cool adiabatically
4) system is cooled at constant volume
What are the 4 stages of a Diesel cycle
1) Gas is compressed adiabatically
2) Heat is supplied whilst pressure is constant (increase V)
3) Gas is allowed to cool adiabatically
4) system is cooled at constant volume
What are the differences between real and theoretical engines
- Corners of theoretical indicator diagram are not rounded, as it is assumed that the same air is cycled continuously
- In real engine, volume is not constant during heating, as this could only occur is p and T changes were instantaneous
- Theoretical loop doesn’t include negative work between exhaust and induction, as it assumes same air is cycled again
- Real temperature rise is less than theoretical, as engines have an internal heat source, and the fuel is not completely burned (∴ <max energy released)
- Net work done is greater for theoretical engines, as real engines require energy to overcome friction (∴ inside real loop is smaller)
What is the indicated power of an engine
The net work done by the engine cylinders in 1 second
(∴ = Max theoretical power generated by gases in engine cylinder)
What is the equation for the indicated power of an engine
Indicated power = (Area of loop)x(No. of cycles per second)x(No. of cylinders)
What is the input power of an engine
The amount of heat energy per unit time that could potentially be released from burning fuel
What is the equation for the input power of an engine
Input power = Calorific value x Fuel flow rate
(⇒ Calorific value = How much energy the fuel has stored per unit volume)
What is the output (brake) power of an engine
The useful power output at the crankshaft
What is the equation for the output power of an engine
P = Tω
(rotational power)
What is the friction power of an engine
The power needed to overcome the friction of an engine
What is the equation for the friction power of an engine
Friction power = Indicated power - Output power
What is the thermal efficiency of an engine
Ratio of the energy available from the fuel, to the energy generated by the engine
(Affected by how well energy is transferred into work)
What is the equation for the thermal efficiency of an engine
Thermal efficiency = Indicated power / Input power
What is the mechanical efficiency of an engine
Ratio of the power generated by the cylinders to the power output at the crankshaft
(Affected by the amount of energy lost due to friction between moving parts)
What is the equation for the mechanical efficiency of an engine
Mechanical efficiency = Output power / Indicated power
What is the overall efficiency of an engine
Ratio of the energy available from the fuel, to the power output at the crankshaft
(Accounts for both mechanical and thermal efficiency)
What is the equation for the overall efficiency of an engine
Overall efficiency = Output power / Input power
(∴ = Mechanical efficiency x Thermal efficiency)
What are heat engines
Engines that convert heat energy into work
What is the second law of thermodynamics
It is impossible for heat from a high temperature source to produce an equal amount of work
∴ All heat engines must must operate between a heat source and a heat sink, as some heat must increase the temp of the engine and then be disspursed
(otherwise engine temp would reach source temp and work would stop)
∴ Q(H) = W + Q(C)
Q(H): Heat transfer from the heat source
W: Work done by the engine
Q(C): Heat loss to surroundings (transfer to heat sink)
What is the efficiency of a heat engine
The ratio of the heat transferred from the source, to the useful output work
(Affected by the difference between the source and sink temperature)
What is the equation for the efficiency of a heat engine, relating to the 2nd law of thermodynamics
Efficiency = Q(H)-Q(C) / Q(H)
Efficiency = W/Q(H)
W = Q(H)-Q(C)
∴ Efficiency = Q(H)-Q(C) / Q(H)
What is the maximum theoretical efficiency of a heat engine
T(H)-T(C) / T(H)
Ratio of Q(H):Q(C) = Ratio of T(H):T(C)
Efficiency = Q(H)-Q(C) / Q(H)
∴ Max theoretical Efficiency = T(H)-T(C) / T(H)
as T = max energy transfer
What is the efficiency for real life heat engines less than the maximum theoretical efficiency
- Energy lost to overcome friction
- Fuel doesn’t burn entirely
etc.
How is efficiency maximised in heat engines
To maximise the efficiency of a heat engine, as much input energy as possible must be transferred usefully (∴ Q(C) low as possible)
However, some heat is always lost to the surroundings (Q(C) must be >0), so to avoid wasting energy, some ‘combined heat and power (CHP)’ plants use released heat for other purposes, such as heating local houses, generating electricity, etc.
What is a reversed heat engine
An engine that transfers heat energy from a cold space to a hot space
- Heat naturally flows from hot to cold, ∴ work is required to transfer
- There will always be some energy in the cold space to be transferred, as the temp as above absolute 0
Why are 100% efficient heat engines and reversed heat engines impossible
According to the second law of thermodynamics (Q(H) = W + Q(C)), there must always be a transfer of energy between a hot and cold space.
If 100% of the energy from the hot space was converted into work (or vice versa for reverse engines), the engine temp would eventually be the same as source temp, so no work would be done
T(C) > 0
Max theoretical Efficiency = T(H)-T(C) / T(H)
∴ Efficiency < 1
How do refrigerators work as reversed heat engines
(process of refrigeration)
A refrigerant moves through coils, travelling inside and outside of the fridge:
- The liquid refrigerant outside the fridge is decompressed, decreasing the pressure and therefore decreasing it’s temperature as it flows into the fridge.
- It then begins to absorb energy from it’s surroundings as it travels through the evaporator coils inside the fridge, decreasing the temperature of the surroundings as the refrigerant heats up to become a gas.
- As it leaves the fridge, the gaseous refrigerant is compressed, increasing the pressure, and therefore the temperature.
- It then begins to travel through the condenser coils outside the fridge, where it releases it’s energy to the surroundings, cooling as it returns to a liquid
- The cycle then repeats as energy is absorbed from in the fridge and released to the surroundings, due to the work done during the compression and decompression stages
How do heat pumps work as reversed heat engines
Work similar to refrigerators, as the refrigerant is sequentially compressed and decompressed to transfer energy from the cold space (outside the house), to the hot space (inside the house)
However, for a heat pump:
- Heat given out by compressed refrigerant usually heats a water tank
- A fan will usually amplify the absorption of heat in the cold space
What is the coefficient of performance (COP)
A measure of how well work is transferred into heat energy by a reverse heat engine:
- Ratio of Q:W
- ∴ = 1/efficiency, so shows how well work causes energy transferer, (whereas shows how well energy transfer causes work, in regular heat engines)
- Can be greater than 1
What is the equation for the COP for a refrigerator
COP(ref) = Q(C) / (Q(H) - Q(C))
For a refrigerator, the heat removed from the cold space determines how well it works
∴ COP(ref) = Q(C)/W
W = Q(H) - Q(C)
∴ COP(ref) = Q(C) / (Q(H) - Q(C))
What is the maximum theoretical COP for a refrigerator
At max theoretical efficiency:
COP(ref) = T(C) / (T(H) - T(C))
What is the equation for the COP for a heat pump
COP(hp) = Q(H) / (Q(H) - Q(C))
For a heat pump, the heat transferred to the hot space determines how well it works
∴ COP(hp) = Q(H)/W
W = Q(H) - Q(C)
∴ COP(hp) = Q(H) / (Q(H) - Q(C))
What is the maximum theoretical COP for a heat pump
At max theoretical efficiency:
COP(hp) = T(C) / (T(H) - T(C))
