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ASRJC H2 Physics
Summary Notes 202 3
1 Measurements Summary Notes
A. Physical Quantities and SI Units
- Base quantities and units
Base Quantity
Base Unit Name Abbrev Length metre m Mass kilogram kg Time second s Electric current ampere A Temperature kelvin K Amt of subst mole mol
Avogadro Constant, NA 1 mol of 12 C is 12g,
1 atom of 12 C is approx 12u.
e A
m N
n= =
M N
- Derived units are products or quotients of base units
Rules for working out the units of derived quanitites: i. If 2 or more quantities are added together, they must have the same units. ii. The rules of algebraic multiplication and division apply
- A homogenous equation is one where all the terms in the equation have the same units.
- A physically correct equation is homogeneous, but a homogeneous equation may not be physically correct.
- Useful prefixes (memorise)
pico 10 -^12 p deca 101 da nano 10 -^9 n hecto 102 h micro 10 -^6 kilo 103 k milli 10 -^3 m mega 106 M centi 10 -^2 c giga 109 G deci 10 -^1 d tera 1012 T
B. Errors and Uncertainties
- A systematic error will result in all readings having a constant error in one direction.
- A random error will result in a scatter of readings about a mean value.
- Systematic error vs Random error
Systematic Error Random Error Constant deviation of readings in 1 dir from the true value
Scatter of readings about mean value
Eliminated by calibration curve and control experiments
Reduced by taking average of repeated readings
- Accuracy is the degree to which a measurement approaches the true value.
- Precision is the degree of agreement of repeated measurements of the same quantity.
- Precision vs Accuracy
Precision Accuracy degree of agreement between means of the same quantity
degree of agreement between experiment value and true value Readings close to mean
Mean close to true value Small random error Small systematic error
- Determining the uncertainty in a derived quantity
i. If y = kx + z (k is a const)
then y = k x + z
ii. If 3
2
z
wx
y = k (k is a const)
then
z
z
x
x
w
w
y
y
- Actual uncertainty, y, is always expressed to 1 s.f.
- Fractional (Δy/y) or Percentage uncertainty can be to 2 or 3 s.f. ( only 2 s.f. for practical )
n – no of mol; N – no of atom m – mass; Mr – molar mass
2 Kinematics Summary Notes
A. Rectilinear Motion
- Displacement is the distance of an object or a point, in a specified direction, from some reference point.
- Speed is the distance travelled divided by the time taken.
- Average speed is the total distance travelled
divided by the total time taken.
- Velocity is the rate of change of displacement.
- Average velocity is the total change of
displacement divided by the total time taken.
- Acceleration is the rate of change of velocity.
- Acceleration of free fall is the acceleration of a body towards the surface of Earth when the only force acting on it is its weight.
- Instantaneous vs Average
Instantaneous Average
𝒗 =
Gradient at a point in an s-t graph
total displacement total time
𝒂 =
Gradient at a point in a v-t graph
total change in velocity total time
Graph Gradient Area s-t v -
v-t a s
a-t - v
- Derive the equations of motion (for linear and constant acceleration motion)
v = u + at s = ut + 12 at^2
v^2 = u^2 + 2 as s = 12 ( u + v t )
State sign convention when using the equations of motion
- Graph for free falling object
B. Projectile Motion
1
1
1
dtt t
ds
v
t
s
t
s
v
=
s
t (0,0)
s 1
t 1
s
t
w/o Rair
w Rair
v
t
w/o Rair w Rair
s
t where Rair is air resistance
u
u
u
u x = u
Take as +ve, u y = 0 a y = g
u x = u cos
Take as +ve,
u y = u sin
a y = − g
u x = u cos
Take as +ve,
u y = u sin
a y = g
- To solve projectile motion problems: i. Resolve initial velocity u into horizontal and vertical components ii. Consider the horizontal and vertical motion separately iii. Set a positive convention in both directions (usually take dir of initial velocity to be +ve) iv. Apply the equations of motions
horizontal motion: a x = 0
vertical motion: a y = g (if taking ↓ as +ve)
- Projectile motion with air resistance
- Shorter time
- Shorter range
- Lower maximum height
- Asymetrical
y displacement
w/o Rair
w Rair time
y displacement
w/o Rair
w Rair horiz distance
- An elastic collision is one in which the total
kinetic energy remains the same. Use
✓ COM ✓ Conservation of Kinetic Energy ✓ Relative speed of approach = Relative speed of separation ➔ u 1 – u 2 = – (v 1 – v 2 ) [Note: must include direction]
- An inelastic collision is one in which the total
kinetic energy is not conserved. Use
✓ COM If objects move together after collision, the collision is definitely inelastic.
BUT if objects do not move off together after collision, the collision MAY be or MAY NOT be elastic, question will have to give other information for a conclusion to be made.
- For all collisions, momentum and total energy of the system is always conserved if the system is isolated.
4 Forces Summary Notes
A. Types of force
- Contact force Contact force has two components, namely the normal contact force and the frictional force (see below).
- Resistive force
Frictional force (static and kinetic) and viscous force (force on body when it moves through fluids)
Hooke’s law states that, provided the proportionality limit is not exceeded, the extension of a body is proportional to the applied load.
F = k e, where k is spring constant and e is the extension or compression
- Upthrust is equal in magnitude and opposite in direction to the weight of fluid displaced by a submerged or floating object.
Upthrust = weight of fluid displaced = Vρg,
where V is the volume of fluid displaced and ρ is density of fluid.
For an object floating in equilibrium , the upthrust is equal in magnitude and opposite in direction to the weight of the object.
B. Centre of gravity
- The centre of gravity of an object is the point at which the whole weight of the object may be considered to act.
C. Turning effects of forces
- Moment of a force is the turning effect of the
force. It is equal to the product of the force and the perpendicular distance of the line of action of the force from the pivot.
Moment of T about O = T sin θ x d SI Unit: N m
- A couple consists of two equal and opposite parallel forces whose lines of action do not coincide. It tends to produce rotation only.
- The torque of a couple is the turning effect of the couple. It is equal to the product of one of the forces and the perpendicular distance between the forces.
Torque of a couple, τ = F x d
- Principle of Moments states that for a system in equilibrium, there is no resultant moment about any point.
Total clockwise moments = total anti-clockwise moments
D. Equilibrium of forces
- Conditions for a system to be in equilibrium :
- No resultant force.
- No resultant torque about any point.
- Equilibrium condition results in:
- Closed vector triangle / polygon
- For 3 non-parallel forces in the same plane, the lines of action of the forces must intersect at a common point.
T
O
W
θ
R
d
F
F
d
- Potential energy is defined as the stored energy
of an object to do work as a result of its position or shape.
- Gravitational Potential energy is the stored
ability of an object to do work as a result of its mass and position.
- Electric Potential energy is the stored ability of
object to do work as a result of its charge and position.
- Elastic Potential energy is the stored ability of
object to do work as a result of its shape.
C. Conservation of Energy
- The Principle of conservation of energy states that energy cannot be created or destroyed, but it can be converted from one form to another.
- KEi + PEi = KEf + PEf + Work done against dissipative forces
D. Power & Efficiency
- Power is the work done per unit time.
- Instantaneous power can be determined from the gradient of energy-time graph.
- Derivation of P = Fv
- Efficiency, η= (output / input) x 100%
F
h PE =0^ m
Consider an object of mass m in a uniform gravitational field being lifted up by a force F to height h above the ground with a constant velocity.
From the definition of work, work done by the force F is the product of the force and the displacement in the direction of the force
Work done, W = Force x displacement= F x h
Since magnitude of force F = Weight of object (as object moves at constant velocity) Hence, W = Weight of object x h = mg h
The gravitational potential energy gained is equal to the work done by the force F to move the object to height h.
Therefore, U = W = mgh
Consider a constant driving force F exerted on a body that moves with constant velocity v.
Work done by F in moving a distance d x is given by d W = F d x Hence, power P delivered by F , dW d = = dt dt
F x P^ = Fv
P = Fv
6 Motion in a Circle Summary Notes
1. Kinematics of uniform circular motion
Angular displacement , is the angle through which an object turns, usually measured in radians
(rad). Arc length s Radius r
Angular velocity is defined as the rate of change of angular
displacement.
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
Period T is the time taken by an object to complete one revolution. Frequency F
Tangential velocity v = r
2. Centripetal acceleration
Uniform circular motion – constant speed, velocity continuously changing Centripetal acceleration a always directed towards centre of circle (perpendicular to velocity)
3. Centripetal force
Centripetal force is the resultant of one or more real forces; not an additional force
No work done by centripetal force since it is perpendicular to displacement.
4. Problem solving technique 1. Identify the object moving in circular motion. 2. Identify all forces acting on the object. 3. Find the resultant force towards the centre of the circular motion. 4. Equate this resultant force to the centripetal force expression.
For vertical circular motion, circular motion is not uniform.
Need to apply conservation of energy K.EA + P.EA = K.EB + P.EB
s = r
dt
d
= , unit rad s-^1
T =
T
f
( )^22
r
r
r
a = =
r
v
a
2
2
2 mr
r
mv
Fc = mac = =
Definitions
Field of force is a region of space where a body experiences a force.
Gravitational field is a region of space where a mass experiences a force. The direction of the field is the direction of the force on the mass.
Gravitational field strength at a point is the gravitational force exerted per unit mass placed at that point. Newton’s Law of Gravitation states that the mutual force of attraction between any two point masses is proportional to the product of the masses and inversely proportional to the square of their separation.
Weightlessness is the situation when a person experiences free fall and has zero contact force on him.
[Need to elaborate base on the context of the question. For example: Both the satellite and the astronaut are falling freely with the same acceleration (or the gravitational force on the astronaut is just enough to provide the centripetal force for the circular motion), the force the satellite exerts on the astronaut is zero, hence the astronaut experiences weightlessness. (N05/P2/Q3(c))]
The gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point.
Note: Gravitational Potential is always negative because the gravitational potential is taken to be zero at infinity and gravitational forces are attractive, work done by the external agent on the point mass moving it from infinity is negative.
Geostationary orbit refers to a circular orbit around the Earth in which a satellite would appear stationary to an observer on the Earth’s surface because it revolves around the Earth with the same period and in the same direction as the rotation of the Earth.
Note: A geostationary satellite must be placed vertically above the equator because the force of attraction to the Earth is to its centre so the circular orbit must be centred on the Earth's centre. If the orbit is not on the equator the satellite would have varying latitude and so not be geostationary.
A geostationary satellite must move from west to east because the earth rotates from west to east.
8 Temperature and Ideal Gases Summary Notes
1. Thermal equilibrium
Temperature is a measure of the degree of hotness of an object. Thermal energy is transferred from higher to lower temperature.
Thermal equilibrium is attained when two bodies in thermal contact have the same temperature and there is no net flow of heat between them.
Zeroth Law of Thermodynamics states that if two bodies X and Y are separately in thermal equilibrium with a third body T, then X and Y are in thermal equilibrium with each other.
2. Temperature Scales
(i) For calibrated empirical temperature scale, assume a linear relationship between thermometric property and the temperature.
E.g. the Centigrade scale
However, using different thermometers to measure the temp of an object, the temp readings obtained were different as thermometric property X may not vary linearly with temperature.
Different thermometric properties do not respond in the same way to changes in temperature. They only agree at the two fixed points, by definition. (ii) Absolute Scale of temperature (thermodynamic scale of temperature) is a scale that is independent of the property of any particular substance and has an absolute zero.
- Lower fixed point is absolute zero 0 K at which all substances have minimum internal energy
- Upper fixed point is triple point of water 273.16 K at which all 3 states of water (saturated water vapour, pure water and melting ice) co-exist in equilibrium
- One kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water
(iii) Celsius Temperature Scale
3. Equation of State
PV = nRT
Avogadro constant is the number of atoms in 0.012 kg of carbon-12.
One mole of any substance is the amount containing 6.02 × 10^23 particles.
X = value of thermometric property at unknown temperature ºC
0
Xo
100
X 100
X
Property, X
Temperature/ oC
100 −
= − 100 o
o X X
X^ X
/ 0 C = T / K – 273.
where P = pressure of the gas (SI unit: Pa),
V = volume of the gas (SI unit: m^3 ),
n = number of moles of gas present,
R = molar gas constant = 8.31 J K−^1 mol−^1 (given in the data booklet)
T = temperature of the gas (SI unit: K)
The assumptions of the kinetic theory of gases include:
- A gas consists of a large number of identical molecules.
- The gas molecules are in rapid, random motion.
- There are no inter-molecular forces between the molecules, except during a collision.
- The volume of the gas molecules is negligible compared with the volume of the container.
- Collisions between gas molecules and with the walls of the container are elastic.
- The time taken for a collision is negligible compared with the time interval between collisions.
Pressure due to a Gas
p =
3 ρ c^2 or pV = 1 3 Nm c^2
Derivation
Velocity of molecule v given v^2 = vx^2 + vy^2 + v z^2 …… (1)
Change of momentum of the molecule p = pf - pi = (-mvx) – (mvx) = – 2mvx
Average force acting on a molecule by wall Fm given by Fm = Δt
Δp= −^ 2mvx 2 l vx
2
−mvx
By Newton’s 3rd^ Law, average force on wall by the molecule, F = – Fm =
2
mvx
Total force on the wall F =
Nmvx^2 l ……… (2)
Pressure on the wall p =
Nmvx^2 l^3 ….… (3)
2 z
2 y
2 x c 2 = v + v + v where <c^2 > is mean square speed of gas molecules
2 2 x 3 c
1 v ^ ……… (4)
(^22)
x c
V
Nm
v
V
Nm
p = = ….. (5)
Therefore p =
3 ρ c^2 or pV = 1 3 Nm c^2
6. Kinetic Energy of a Molecule
Using = ^2 3
(^) pV^1 Nm c and pV = NkT ,
Average translational kinetic energy of one molecule: m c kT 2
3 2
(^1 ) =
9 First Law of Thermodynamics Summary Notes
1. Specific heat capacity and specific latent heat
Thermal energy is the energy transferred by conduction, convection or radiation from one body to another due to a temperature difference.
Heat capacity, C , is the amount of thermal energy Q required per unit change in temperature of the substance. SI unit of C is J K-^1
Specific heat capacity , c , is the thermal energy Q per unit mass per unit change in
temperature of the substance. SI unit is J kg-^1 K-^1 where = f – i
Specific latent heat of vaporization , Lv , is the thermal energy Q per unit mass required to change a substance from liquid to vapour without change of temperature. SI unit of Lv is J kg-^1.
Specific latent heat of fusion , Lf , is the thermal energy Q per unit mass required to change a substance from solid to liquid without change of temperature. SI unit of Lf is J kg-^1.
Note: Energy gain or loss needs to be considered for setups measuring latent heat. To do that, measurements are done with same setup using same time period and temperature or range of temperature but with different power. Heat loss depends on temp difference between substance involved and surrounding temperature, and time period.
2. Internal Energy
The internal energy is the sum of random distribution of kinetic and potential energies associated with the molecules of a system. i.e. U = Ek + Ep
Raising temperature increases Ek ; changing the state changes Ep
3. First Law of Thermodynamics First Law of Thermodynamics states that the increase in internal energy Δ U of a system is equal to the sum of the heat supplied Q to the system and the work done W on the system.
U = Q + W
Δ U = increase in internal energy of a system (Δ U = U f – U i )
Q = heat supplied to the system
W = work done on the system where W^ =− p V or area under p-V curve
Q = C
Q = mc
Q = mL
a) Displacement x is the linear distance of the oscillating body from its equilibrium position in a specified direction.
Amplitude xo is the maximum displacement of an oscillating body from its equilibrium position.
Period T is the time taken to complete one oscillation.
Frequency f is the number of oscillations per unit time.
Angular frequency is a constant of a given oscillator and is related to its natural frequency f by = 2 f.
= 2 f ,
f
T
b) SHM : acceleration a = - ^2 x ( Defining equation ) x = xo sin t , v = vo cos t , or x = xo cos t , v = - vo sin t , v = ( xo^2 – x^2 )
Simple harmonic motion is the motion of a body such that its acceleration is proportional to its displacement from the equilibrium position and is always directed towards that point.
c) Graphical illustrations of x, v, a for SHM:
d) Interchange between kinetic and potential energy (with respect to displacement):
e) Damping is the dissipation of total energy of an oscillating system with time due to resistive forces.
Damped oscillations: light, heavy, critical damping.
f) Damped oscillations are oscillations in which the amplitude diminishes with time as a result of resistive forces that reduce the total energy of the oscillations.
g) Critical damping occurs when the displacement of the body is reduced to zero in the minimum time possible without any oscillations occurring.
h) Forced oscillation is produced when a body is acted upon by an external periodic driving force, causing the body to oscillate at the driving frequency, rather than the natural frequency of the body.
i) Resonance occurs when the driving frequency of a body is equal to its natural frequency, giving a maximum amplitude of oscillation.
The frequency response and the sharpness of the resonance are determined by the degree of damping as shown in the graph.
j) Circumstances where resonance is useful: Sound box of musical instruments, microwave oven, radio aerial
k) Circumstances where resonance should be avoided : Car vibrates at certain engine speed, Tacoma bridge
10 Oscillations Summary Notes
wrt t (start@equilibrium)
wrt x max values
x xosin( t) - xo
v xocos( t) x o^2 − x^2 vo = xo
a ^2 xosin( t) ^2 x ao = ^2 xo
KE ½ m ^2 xo^2 cos^2 ( t) ½ m ^2 (xo^2 − x^2 ) ½ m ^2 xo^2
If PE = 0 at equilibrium position
PE ½ m ^2 xo^2 [1-cos^2 ( t)] ½ m ^2 x^2 ½ m ^2 xo^2
TE ½ m ^2 xo^2 ½ m ^2 xo^2 ½ m ^2 xo^2
Note: TE = KEmax = PEmax
EPE
KE
GPE
Total Energy
Energy
Example of Energy-position diagram where components of PE (e.g. GPE and EPE) are separately represented
Notes:
- Total Energy = Total PE + KE or (EPE + GPE) + KE
- The reference point for GPE component can be adjusted such that at equilibrium, Total PE = 0 (hence KEmax = TE still valid)
Total PE
Lowest point
Equilibrium (^) Highest point
