physics notes for j1
DESCRIPTION
credits to RayTRANSCRIPT
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 1 of 34
Foreword
I feel that amongst all Singapore students, many of us may not have the privilege of receiving quality education in the subject of physics due to differing teaching pedagogies used by various institutions and teachers/mentors.
Despite my limited ability, I hope that these notes will assist you in your learning journey for physics, be it the ‘A’ you are aiming for, or to sustain your genuine interest in the subject.
Unlike many other subjects, physics has
apparently been one where many misconceptions
arise. Furthermore, being an applied subject, it is
one where memorizing gets you the ‘U’. It is the
understanding, deduction and math that count.
I do hope you see where physics is around you in
this world. From your air-conditioners to cars to
infrastructure, physics is everywhere. If you can
learn to appreciate the greatness of mankind’s
inventions, surely you can appreciate the beauty
of physics.
With that understanding, I wish you all the best for H2 Physics for your promotional exams.
Ang Ray Yan
Hwa Chong Institution (11S7B)
Disclaimers / Terms and Conditions
- Physics needs tons of practice. This note gets you the ‘U’ grade if you only read it.
- g on Earth is defined as 9.81ms-2 unless specified otherwise.
- There might be errors. Please use some discretion when reading through. This note is definitely not the best.
- Definitions are given in boxes
- Even at A’ levels, due to the nature of the subject, only concepts appear here.
- Drawing and graphs are equally important in terms of scoring. It is after all, representations and interpretations of our real world.
- The use of the any calculator is not covered in this note. It is assumed that you have prior knowledge on its use.
- I don’t believe in strong O’ level concepts because I learnt little in my high school years. This note tries to include the basics.
- All content in this set of notes may or may not be accurate in the real world since most of it comprises classical mechanics.
- These notes serve main as a concept check. Applying the concepts is another issue.
- Distribute only to students by email or thumbdrive. The usage of these notes by any school or tuition teacher is strictly prohibited.
- This is meant for J1 students only. I strongly recommend all J2 students to practice on problems instead of wasting time here.
- If you bought a copy of this, please ask for a refund. It is free!
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 2 of 34
Contents Page
Measurements 3r-6r
- Physical Quantities and Units 3r-4r
- Errors and Uncertainties 4r-5l
- Precision and Accuracy 5r
- Random and Systematic Errors 5r
- Scalars and Vectors 6l-6r
- Rules of Significant Figures 6r
- Homogeneous Equation 6r
- Dimensional Analysis 6r
Kinematics 7l-9r
- Terminologies 7l
- Describing Motion with Diagrams 7l-7r
- Describing Motion with Graphs 7r
- Equations of Motion 8l
- Free-fall Bodies 8l
- Drag Force 8r
- Projectile Motion 8r-9r
Dynamics 9r-14l
- Types of Forces 9r-10l
- Newton’s 3 Laws 10l-11l
- Conservation of Linear Momentum 11r-12r
- Collisions 12r-13r
- Coefficient of Restitution 13r
- Static and Kinetic Friction 14l
Forces 14l-15r
- Hooke’s Law 14l
- Upthrust / Buoyant Force 14l-14r
- Translational Equilibrium 15l
- Moments 15l
- Rotational Equilibrium 15l
- Static and Dynamic Equilibrium 15l
- Three-force Systems 15r
- Couples 15r
Work, Energy, Power 14r-16l
- Definitions 14r
- Work done 15l
- Mechanical Energy 15l-15r
- Conservation of Energy 16l
- Power and Efficiency 16l
Circular Motion 16r-18l
- Kinematics of Circular Motion 16r-17l
- Uniform Circular Motion 17l
- Centripetal Acceleration / Force 17l-17r
- Vertical Circular Motion 17r-18l
Gravitation 18l-21r
- Law of Universal Gravitation 18l-19l
- Geostationary Satellites 19l
- Gravitational Field Strength 19r-20l
- Weightlessness 20l
- Gravitational Potential 20l
- Gravitational Potential Energy 20r-21l
- Escape Speed 21r
- Binary Star System 21r
Oscillations 22l-24l
- Introduction 22l
- Simple harmonic Motion (S.H.M) 22l-23l
- Damping 23l-23r
- Resonance 23r-24l
Waves 24r-27l
- Introduction (Terms and Graphs) 24r
- Wave Equation 24r-25l
- Transverse vs. Longitudinal Waves 25l-25r
- Phase Difference 26l
- Electromagnetic Waves 26l-26r
- Intensity of Waves 26r
- Polarization 27l
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 3 of 34
Superposition 27r
- Principle of Superposition 27r
- Interference 27r-28l
- Diffraction and
Huygen’s-Fresnel Principle 28l-28r
- Young’s Double Slit Experiment 28r-29l
- Diffraction Grating 29r-30l
- Stationary Waves 30l-30r
- Stretched Strings 30r-31l
- Air Columns 31l-31r
Miscellaneous 33l-34l
- Useful Knowledge / Summary 33l-34l
Credits 34l
Chapter 1: Measurements
1.1 Physical Quantities and Units
Physical quantities are properties that can be measured/calculated and expressed in numbers.
1.1.1 International System of Units (SI)
Established in 1960 by the 11th General
Conference on Weights and Measures, the
following are the 7 SI Base Quantities and Units:
Base Quantity Base Unit Symbol
length metre m
Length of path travelled by light in vacuum during a time interval of 1/(299,792,458) of a second.
Base Quantity Base Unit Symbol
Mass kilogram kg
Mass of the international prototype of kilogram (made of platinum-iridium, kept at BIPM)
Base Quantity Base Unit Symbol
Time second s
Duration of 9,192,631,770 periods of the radiation corresponding to the transition between 2 hyperfine levels of the ground state of the Casesium 1333 atom.
Base Quantity Base Unit Symbol
Electric Current Ampere A
The constant current which, if maintained in 2 straight, parallel, 1m apart conductors of infinite length and negligible circular cross section, would produce between the conductors a force equal to 2 x 10-7 Nm-1.
Base Quantity Base Unit Symbol
Thermodynamic Temperature
Kelvin K
It is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
Base Quantity Base Unit Symbol
Amount of Substance
mole mol
The amount of substance in a system containing as many elementary entities as there are atoms in 0.012kg of carbon-12.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 4 of 34
Base Quantity Base Unit Symbol
Luminous Intensity
Candela cd
Candela is the luminous intensity in a given direction of a source emitting monochromatic radiation of frequency 540 x 1012 Hz having a radiant intensity in that direction of 1/683 Wsr-1
From the base quantities and units, we can obtain
derived quantities and units:
Quantity Formula Units Usual Units
Volume - -
Density
-
Velocity
-
Acceleration
-
Force
Momentum Pressure
Energy - Moment -
Power
Electric Charge
Voltage
V
Resistance
Frequency
1.1.2 Prefixes (Common Ones only)
Factor Prefix Symbol
pico p
nano n
micro
milli M
centi C
deci D
- -
kilo K
mega M
giga G
tera T
1.1.3 Rapid Estimation
Known as ‘Fermi’ problems, estimation uses
simple numbers (e.g. 2, 5) with the correct order
of magnitude (e.g. 10-3 or 104):
This goes by a 3-step process:
First, identify the unknown:
Next, identify the known:
Lastly, find relation between known and unknown:
Note: Actual area of Singapore is 682.7km2.
1.2 Errors and Uncertainties
The experimental error in measuring a physical quantity can be interpreted as the difference in the measured and true value of it.
Do note that we usually do not know what the
true value is. (Hence the need to measure)
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
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1.2.1 Absolute Uncertainty
Hence, we express values as such:
Where is the measured value and is the
estimated certainty. In this case, is the
absolute uncertainty of .
Do note that each reading taken has its own
estimated uncertainty.
Do note that all absolute uncertainties should
have 1 significant figure and should have the
same decimal places as .
1.2.2 Fractional / Percentage Uncertainty
1.2.3 Combining Uncertainties
| |
For any other functions, we calculate as follows:
Examples of Z include sine, cosine, and any
function with a maximum and minimum.
1.3 Precision and Accuracy
Precision refers to the closeness of a set of measurements.
Accuracy refers to the agreement between the measured and true value of a quantity.
The following illustrates the idea:
Target Accurate Precise
no yes
yes no
yes yes
no no
1.4 Random and Systematic Errors
Random errors occur as a scattering of readings about the average value of measurements. They have varying signs and magnitudes. It can only be reduced by combining measurements (e.g. thickness of 100 A4 paper, not 1) or by repeating measurements and taking averages.
Systematic errors occur as a shift of value from the true value of measurements. They have similar signs and magnitudes. It can eliminated by accounting for it (e.g. zero errors in instruments, calibration etc.)
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
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1.5 Scalars and Vectors
Scalars only have magnitude.
Vectors have magnitude and direction.
The parallelogram law of addition (left)
demonstrates how to sum 2 vectors, and the
polygon law of addition shows how to sum all
vectors, giving the resultant vector. Also,
| |
1.5.1 Finding Resultant Vectors
To find the resultant vector (green):
(
)
1.5.2 Resolution of Vectors
As shown above, a vector can be resolved into the
vertical and horizontal components. This is to
ensure that they the components are
perpendicular and independent of each other.
Usually, rightwards and upwards is positive and
leftwards and downwards is negative. However,
the question definition takes priority.
1.6 Rules of Significant Figures
This is generally summarized into 3 rules:
For multiplication/division, use least s.f. for result.
For addition/subtraction, use least d.p. for result. For logarithms, the number of s.f. we take logarithms is the number of d.p. for the solution:
1.7 Homogeneous Equation*
Homogeneous equations are equations where units on LHS=RHS.
There are 2 reasons why a homogeneous
equation may not be physically correct:
Coefficient
Missing terms
1.8 Dimensional Analysis*
Dimensions correspond directly with base units.
For instance:
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
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Chapter 2: Kinematics
Kinematics (Greek: ) is the branch of
mechanics that describe motion.
2.1 Terminologies
Distance [x] (red) is the total length travelled by a moving object irrespective of direction of motion.
The displacement [s] (black) from a reference point, O, is the linear distance and direction of the object from O.
The speed of an object is the rate of change of distance travelled by an object with respect to time.
The velocity of an object is the rate of change of displacement with respect to time.
The acceleration of an object is the rate of change of velocity with respect to time.
∫ ∫
2.2 Describing Motion with Diagrams
2.2.1 Ticker Tape Diagrams
A ticker places a tick on tape dragged by a moving
object. The distance between dots represents the
object’s position change during a defined time
interval (e.g. 0.1s). Hence, we see the top object
travels at a constant speed, whilst the bottom is
accelerating.
2.2.2 Vector Diagrams
Vector arrows are used to depict direction and
relative magnitudes of an object’s velocity. Thus,
we see the top object travels at a constant speed,
whilst the bottom is accelerating.
2.2.3 Stroboscopic Photographs
Stroboscopic photographs are photos taken by
cameras with an open shutter. With a flashing
light at fixed frequencies, a fixed duration
between illuminations produces the different
positions of the object.
2.3 Describing Motion with Graphs
2.3.1 Displacement Time Graphs (s-t)
The gradient of the graph
gives the instantaneous
velocity (reddish-brown
at ). The slope of
connecting line gives the
average velocity. (green
between and ).
2.3.2 Velocity Time Graphs (v-t)
The gradient of the graph
gives the instantaneous
acceleration (reddish-
brown at ). The slope of
connecting line gives the
average acceleration.
(green between and ). The area under the
graph gives the displacement.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 8 of 34
2.4 Equations of Motion
All equations here assume a constant
acceleration.
Hence, we can deduce the equation of velocity:
∫
Finally, we can integrate to get the displacement:
∫
To express displacement in terms of velocity,
2.5 Free-fall Bodies
Freely falling objects is any object moving only under the influence of gravity (i.e. ignore air resistance etc.). They accelerate downwards at 9.81ms-2
2.6 Drag Force
When a body moves through liquid or gas, a drag
force is experienced. It depends on the velocity of
the body. Other factors include shape and
dimension of the body and the viscosity of liquid.
Source:http://www.equipmentexplained.com/images/physics_image
s/fluid_images/flow_images/basics/laminar_turbulent_flow.gif
For Laminar flow (low velocity), the drag force (FD)
is given by:
For turbulent flow (high velocity), the drag force
(FD) is given by:
Hence, applying it to air resistance, we have the
following:
For a body in free fall with air resistance,
the drag force will increase until it is
equal to the weight. Since the net force
will be zero, the object reaches terminal
velocity:
2.7 Projectile Motion
Using the resolution of vectors, we know that the
horiztonal and vertical motions are independent
of each other.
Using the equations of motion,
Horizontal Vertical
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 9 of 34
Hence, the impact velocity is given by:
√
2.7.1 Maximum Height (H)
At maximum height,
2.7.2 Duration of Flight (tflight)
Assuming projectile lands on a level ground as it is
initially fired at:
2.7.3 Horizontal Range (R)
For horizontal motion, we know that:
Hence, to get maximum range,
2.7.4 Trajectory Equation
Trajectories are parabolic, as proven below
(
)
(
)
(
)
2.7.5 Projectile Motion (with Air Resistance)
On the flight upwards, air resistance acts in the
same direction as weight, hence the maximum
height is lowered (total downward force larger).
On the flight downwards, air resistance acts in the
opposite direction as weight. Hence, time to
travel up is greater than time to travel down.
Also, note that the path is asymmetrical and the
horizontal range is lower.
Chapter 3: Dynamics
This topic studies the cause of motion and
changes in motion due to forces.
3.1 Types of Forces
There are contact forces and non-contact forces.
Contact forces are in physical contact.
Contact forces are in not in physical contact, and act at a distance.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 10 of 34
3.1.1 Centre of Gravity
The centre of gravity (cg) is the point at which the weight of an object appears to act on.
Suppose we have particles
denoted by and their
mass is , then it follows
that:
∑
∑
3.1.2 Contact Force and Friction
The normal contact force is due to electrostatic
repulsion between molecules of the surface and
the object. It balances the weight, directed
perpendicular to the surface.
Friction always acts in the opposite direction of
relative motion. It will be discussed further in 3.6.
3.2 Newton’s 3 Laws
3.2.1 Newton’s First Law
Newton’s First law states that a body stays at rest or continues to move with a constant speed in a straight line unless a net external force acts on it.
An object’s resistance to change in its state of
motion is known as inertia. Note that the larger
the mass, the higher the inertia.
3.2.2 Newton’s Second Law
First we must understand linear momentum.
The linear momentum of a body is the product of its mass and velocity.
Hence, we can now define the following:
Newton’s second law of motion states that the rate of change of linear momentum is in the same direction and directly proportional to the resultant force acting on it.
Note that if we differentiate it,
Lastly, we define a new term:
The impulse a force is the product of the force and the time interval over which it is applied.
Impulse is the area under a force-time graph. The
average force is represented by a rectangle (e.g.
green, left) Do note that large force applied over a
short time (yellow) hence the same impulse as
small force applied over a long time (green, right).
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 11 of 34
We also have the impulse-momentum theorem:
∫ ∫
∫ ∫
∫
To effectively solve problems, the usage of free
body diagrams (below) is crucial. Thus, we label
all forces acting on an object (the block).
Resolving vectors might be necessary.
3.2.3 Newton’s Third Law
Newton’s third law states that if body A exerts a force on body B, body B will exert an equal and opposite force of the same nature on body A. Note: both forces must act on different bodies
Also, note that for all connected components (be
it by string, contact, etc.), they have the same
acceleration:
(
)
3.3 Conservation of Linear Momentum
During collisions, we observe that forces act on
opposite bodies without external forces. (e.g. for
2 billard balls as shown above).
∫ ∫
∑
∑ ∑ ∑
The principle of conservation of linear momentum (PCOM) states that the total linear momentum of a system is conserved if no net external force acts on the system.
∑ ∑
3.4 Collisions
3.4.1 Head-on Collision vs Glancing Collision
A collision is an isolated event where 2 or more colliding bodies exert relatively strong forces on each other for a relatively short time.
For head-on collisions, the direction of motion of
both bodies before and after collision is in the
same line of motion.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 12 of 34
3.4.2 Elastic and Inelastic Collision
An elastic collision is when 100% of kinetic energy
is being conserved.
A completely inelastic collision is when 2 objects
coalesce (stick together) and move with common
velocity after collision. It represents the
maximum possible loss of KE (not loss of all KE).
3.4.3 Relative speed of Approach / Separation
For an elastic 2-body head on collision,
Since KE is 100% conserved for elastic collisions,
That is, for elastic collision, the relative speed of
approach [RSOA] (LHS) equals the relative speed
of separation [RSOS] (RHS).
If 1 of the bodies is initially at rest, then:
3.4.4 Solving Collision Problems
To solve problems, we use PCOM and RSOA/RSOS.
Let A and B be 1.0kg and 3.0kg respectively.
For elastic collision,
For completely inelastic collision,
3.5 Coefficient of Restitution*
The elasticity of a collision is quantified by the
coefficient of restitution,
| |
| |
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 13 of 34
3.6 Static and Kinetic Friction*
Static friction is the force opposing motion
between 2 bodies at rest relative to each other.
Kinetic friction is the force opposing motion
between 2 bodies moving relative to each other.
Hence, there is no static friction when there is
kinetic friction, and vice versa.
Chapter 4: Forces
4.1 Hooke’s Law
Hooke’s law states that the magnitude of the force F exerted by a spring on a body attached to the spring is proportional to the extension x of the spring from equilibrium provided the proportionality limit of the spring is not exceeded.
∫
4.2 Upthrust / Buoyant Force
To understand upthrust, we must first know the
pressure exerted by a fluid.
Using the fluid force acting
on the surface bottom that
offsets the weight of the
water column (dark blue),
Source:http://images.tutorvista.co
m/content/fluids-pressure/liquid-
pressure.gif
Note that the pressure
of fluid acts in all
directions.
Source:http://img.sparknotes.com/fig
ures/0/0a1c01f07d0a0e51105b2065c1
36cda0/ideal_p1_3.gif
The left diagram shows
typical mercury
manometers, measuring
the difference in pressure.
For atmospheric pressure, it
is usually at 760 mmHg. The deeper down the
tube, the higher the pressure (due to extra weight
of column of mercury). in this case gives us the
difference in pressure for 2 gases.
Now, we move on to understand upthrust:
Upthrust is the net upward force exerted by a fluid on a body fully or partially submerged in the fluid.
Hence, to find the upthrust acting on a cube (dark
blue here):
This is Archimedes’ Principle, stating that a body
submerged in liquid has an upthrust equal to the
weight of fluid displaced.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 14 of 34
4.3 Translational Equilibrium
When a body is either stationary or moving at
constant velocity, the body is in translational
equilibrium. The condition is that:
∑
4.4 Moments
The moment of a force about a point is the product of the magnitude of the force and the perpendicular distance of the line of action of the force to the point.
With the example of trying to open a door, we see
that the moment about the hinge is given by:
4.5 Rotational Equilibrium
For a body to be at rotational equilibrium, the net
moment of the body about any point is zero, i.e.:
∑
The principle of moments states that for a body to be in rotational equilibrium,
∑ ∑
must be true for any point on the body.
4.6 Static and Dynamic Equilibrium
Static Dynamic
∑ ∑
e.g. Hanging Picture
e.g. Sliding Ice Block
4.7 Three-force Systems
For stationary bodies experiencing only 3 co-
planar forces, then the lines of action of all 3
forces must intersect at 1 point. (net about that
point must be zero).
Hence, we see that for the bridge to be stable
(suspended by the rope), the direction of force
acting on the bridge by the hinge must meet the
intersection of the other 2 forces.
4.8 Couples
A couple is a pair forces equal in magnitude but opposite in direction whose lines of action are parallel but separate.
Couples only produce rotation and no translation.
The resultant torque is given by:
(
) (
)
Chapter 5: Work, Energy, Power
5.1 Definitions
Work is the transfer and transformation of energy between one body and another.
The energy of a system is a measure of its capacity to do work.
Similar to using momentum and impulse, we can
find the change in energy using work done
without knowing the time interval when the
force is applied.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 15 of 34
5.2 Work Done
The work done on a body is the product of the force and its displacement in the direction of the force.
Hence, given the following diagram,
We conclude that:
Hence, negative work done is doing work in the
opposite direction of displacement. (In the above
case, it could be work done by friction).
Note that the total work done is the area under
the force displacement graph:
∫
For an expanding gas, do note that there is
another formula for the work done:
Provided pressure is constant during expansion,
the force exerted on the piston is constant:
5.3 Mechanical Energy
The total mechanic energy of a system is the sum
of kinetic and potential energy in the system:
5.3.1 Kinetic Energy
Kinetic energy of a body is a measure of energy possessed by the body by virtue of its motion
Using the Newton’s 2nd Law and kinematics
equation for uniform acceleration:
(
) (
)
Hence, a decrease in KE is negative work done,
and the increase in KE is positive work done.
5.3.2 Potential Energy
Potential energy of a body can be defined as the amount of work done on it to give it the current position it occupies.
For an object to exist at its current
position, it needs to overcome the
earth’s attraction:
This is also the gravitational potential energy
(G.P.E) since the object is in a gravitational field.
For an object falling through a distance of :
Hence in general for non-uniform fields
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
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5.4 Conservation of Energy
The principle of conservation of energy states that energy is a quantity that can be converted from one form to another but cannot be created or destroyed. The total energy of an isolated system is constant.
It is an effective method for dealing with various
problems in mechanics, for instance:
To find the maximum compression of spring,
√
√
5.5 Power and Efficiency
Power is defined as the rate of work done.
Also note the following relation with velocity:
(
)
Efficiency is the ratio of useful output power to total input power, i.e.:
The efficiency is usually less than 1 since the input
energy is converted to other non-useful forms of
energy (e.g. heat energy in light bulbs).
Chapter 6: Circular Motion
6.1 Kinematics of Circular Motion
6.1.1 Circular Measure
Given this diagram, we know that
the arc length (red) is given by:
Hence, the radian is defined as the ratio between
arc length and the radius of the circle.
(
)
6.1.2 Angular Displacement and Velocity
If an object moves from to ,
then the angular displacement is
the change in angle ( ).
So similarly, to find angular velocity,
6.1.3 Tangential Speed
Knowing that , we
differentiate w.r.t time:
(
)
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
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6.1.4 Period and Frequency
Period is the time for 1 complete cycle (or revolution)
Frequency is the number of revolutions per unit time.
6.2 Uniform Circular Motion
For a uniform circular motion,
the tangential speed remains
constant, but the direction of
velocity is always changing.
6.3 Centripetal Acceleration / Force
With changing direction and same speed, there
must be acceleration perpendicular to the
velocity vector, known as the centripetal
acceleration (ac). To derive it,
Since the position and velocity vectors move in
tandem, they go around the circle in the same
time, equal to the distance travelled divided by
the velocity:
| |
| |
| |
| |
By equating both equations, we get:
| |
| |
| |
| |
Hence, using Newton’s 2nd Law, the centripetal
force is given by:
Source:http://www.borzov.net/Pilot/FSWeb/Lessons/Student/image
s/Lesson2Figure01.gif
The above shows the banking of a plane, where
tilting the plane gives the horizontal component
of lift responsible for turning (centripetal force).
(
)
6.4 Vertical Circular Motion
When dealing with vertical circular motion, the
conservation of energy becomes very useful:
For instance, if the roller coaster (blue) and its
passengers are 170kg, is travelling at 33ms-1 and
the loop is of radius 19m, we can determine the
normal contact force at the top and bottom and
minimum speed for the roller coaster to pass the
loop safely at the top.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 18 of 34
(
)
(
)
√ √
Chapter 7: Gravitation
7.1 Law of Universal Gravitation
Newton’s Law of universal gravitation states that every particle attracts every other particle with a force directly proportional to their masses, and inversely proportional to the square of the distance between them, i.e.:
Note: Particles are point masses and of negligible dimensions. Objects with radial symmetry can also be treated as a point mass. (Shell theorem). G is the gravitational constant, experimentally determined to be 6.67 x 10-11 N m2 kg-2
7.1.1 Weighing the Earth
Since the moon orbits the moon, we can weigh
the earth using this law and circular motion.
(
)
The same technique applies for the Sun, satellites,
moons and various objects in space. Note that
this is only an estimation.
7.1.2 Acceleration of the Earth
Using the law, we know that the earth would
accelerate towards the apple. Using Newton’s d
2nd 3rd Law,
Hence, we see that the Earth has negligible
acceleration due to its large mass.
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7.1.3 Inverse Square Relationship
Since from the previous example we know that:
(
)
Kepler’s 3rd Law helps to explain how the inverse
square relationship is derived:
(
)
(
)
7.2 Geostationary Satellites
Geostationary satellites are satellites with orbits such that they are always positioned over the same geographical spot on Earth.
Note that it must be in the
same plane as the equator
such that the orbit’s centre
and centre of the Earth is
concentric.
Assuming a circular orbit and that the radius of
the earth is 6.58 x 106m and the mass of the earth
to be 5.98 x 1024 kg,
(
)
√
With such high altitudes, the whole Earth disk is
viewable, but the spatial resolution (amount of
details) is poor. Places further away from the
equator have poorer resolutions.
7.3 Gravitational Field Strength
The gravitational field strength at a point is defined as the gravitational force per unit mass acting on a small mass placed at the point.
If a gravitational field is set up around M and
attracts m which is distance r away from M,
For any spherical body, the acceleration inside it
is zero. For these situations,
At P, the gravitational field
due to solid spherical shell A
is zero. However, the
gravitational field at P due to
spherical mass B (dotted) is
given by:
However, since B is a mass in the shell,
(
) (
)
Hence, if we were to sketch g against r,
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 20 of 34
Note that all gravitational field lines are
perpendicular to the gravitational field vector g.
Also, when g is large, the gravitational field lines
are closer.
However, it is important to note that g is not
uniform on earth. First, the earth is an imperfect
sphere. Since we know that:
Also, note that the density of the earth is not
uniform. With the earth rotating, the
gravitational pull has to also provide for the
centripetal acceleration, lowering g.
7.4 Weightlessness
We know that the weighing balance measures the
normal contact force acting on the object. Hence,
there are 2 types of weightlessness.
True weightlessness is when there is no net gravitational force acting on an object.
We realize that by Newton’s 2nd
Law,
However in this case, since , . Thus,
the reading on the weight machine is zero.
Apparent weightlessness is observed when an object exerts no contact force on its support.
7.5 Gravitational Potential
The gravitational potential at a point in a gravitational field is the work done per unit mass by an external force, in bringing the mass from infinity to that point.
Note: Points of equal distance away from the centre of the Earth are equipotential.
7.6 Gravitational Potential Energy
Assume that point A is infinity, then to move mass
from point A to point B:
At infinity, the gravitational potential energy is 0,
The gravitational potential energy (G.P.E.) of a mass at a point in a gravitational field is the work done by an external force in bringing the mass from infinity to that point.
Since increasing separation distance results in a
gain in G.P.E (gravitational force is attractive in
nature), and infinity is the reference point (U=0),
hence G.P.E is always negative.
7.6.1 G.P.E of a system
To find the number of G.P.E. of
a system with n masses, we
have the following:
For 3 mass, we see that:
This represents the G.P.E. between every 2 point
masses. Hence, for n masses,
∑( ∑
)
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 21 of 34
7.6.2 G.P.E. near Earth’s Surface
At the Earth’s surface, the change in G.P.E is:
(
)
7.6.3 Relationship between G.P.E. and Fg
To move a point further from mass M,
∫ ∫
7.6.4 Total Energy
For any mass m (e.g. satellites) moving in circular
orbit around spherical M, the total energy is:
(
)
7.7 Escape Speed
The escape speed is the minimum speed to project a mass to escape a gravitational field.
√
√ (
) √
7.8 Binary Star System
A binary star system
contains 2 stars. We
know that the force
acting on each other
is:
By using circular
motion, we can
equate them:
(
)
(
)
An object at P experiences true weightlessness.
(g=0 as shown from the . Hence, to
reach from , we only need K.E. sufficient to
reach P:
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 22 of 34
Chapter 8: Oscillations
8.1 Introduction
Oscillation is the repetitive variation of some measure about a point of equilibrium or 2 or more different states.
Free oscillations are systems oscillating at the natural frequency of the system, the frequency characteristic of the system.
Here are some
examples of free
oscillations. In the real
world, they are
subjected to dissipative
forces, known as the
damping effect.
8.2 Simple harmonic Motion (S.H.M)
Assuming we have a particle vibrating along the
lines of XY and the displacement is recorded to
the right. (In a displacement-time graph)
Observing the above, we make some observations
using trigonometry:
Hence, given this generic displacement equation,
we can begin to work out the rest.
8.2.1 Equations in S.H.M
The above graph plots against
Hence, we can now define simple harmonic
motion (S.H.M):
Simple harmonic motion is a periodic motion where an oscillator is subjected to a restoring force directed towards the equilibrium point.
Also, note that to express :
(
)
(
)
√
Hence, we have the following graph (v against x):
√
Note: the red graph ‘moves’ in the
clockwise direction (think about the motion) [For
instance, when , the next moment must
have ]
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 23 of 34
8.2.2 Energy in S.H.M
Given the previous mentioned equations, we can
derive the energy in the oscillator:
Plotting energy against time gives the following:
In essence, it is a sine squared graph (for P.E.)
and cosine squared graph (for K.E.).
8.3 Damping
The progressive decrease in amplitude of any oscillatory motion caused by dissipative forces is also known as damping.
Examples include attaching cardboard (for more
air resistance), immersing oscillators in fluids
(more viscous) and eddy currents.
8.3.1 Light Damping
Oscillating under resistive forces, the amplitude
decreases by the same proportion after each
cycle. Note that the period is slightly longer than
that of the ‘undamped’ value.
8.3.2 Critical Damping
Larger resistive force results in critical damping,
where the oscillator returns to the equilibrium
point in the shortest time without overshooting.
This is used is balances, ammeters/voltmeters to
indicate readings in the shortest time.
8.3.3 Heavy Damping
An even stronger damping force will cause the
oscillator to take a longer time to reach
equilibrium. For instance, over-damped car fuel
gauge indicators are used to give reasonable
indications despite car movement.
8.4 Resonance
Firstly, we must know what forced oscillations are:
Forced oscillations are oscillations under the influence of an external periodic force with a driving frequency.
Next, we move on to investigate resonance.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 24 of 34
Resonance is the phenomenon in which an oscillatory system responds with maximum amplitude to an external periodic force when the driving frequency equals natural frequency of the driven system.
The graph shows various degrees of damping
(light, heavier, and even heavier). Hence, we see
that amplitude of lightly damped systems is very
large at resonance. Damping lowers resonant
frequency to below natural frequency.
With increasing damping, we realize that:
1) The amplitude of oscillation decreases
2) The Resonance peak becomes broader
3) The resonance peak shifts leftwards
4) The graph does not cut at 0. This is because
driving frequency of 0 means there is 1 swing.
Examples of useful resonance include:
1) Microwave cooking (microwave frequency is
natural frequency of water), cooking food
without heating plastic containers too much.
2) Magnetic Resonance Imaging (MRI) allows
analysis of energy absorption using strong EM
fields to produce images (similar to X-rays).
Examples of destructive resonance include:
1) When an opera singer projects a high-pitched
note matching the natural frequency of glass,
glass vibrates at large amplitudes, breaking it.
2) Collapse of bridges (e.g. Tacoma Narrows
suspension bridge). High winds results in
resonance. Hence, the bridge vibrates at
exceptionally large amplitudes and collapses.
Chapter 9: Waves
9.1 Introduction (Terms and Graphs)
Wave is a disturbance of some physical quantity. As the disturbance propagates through space or medium, energy and momentum can be transferred from 1 region to another.
Source:http://rpmedia.ask.com/ts?u=/wikipedia/commons/thumb/7
/77/Waveforms.svg/350px-Waveforms.svg.png
Waves can come in many waveforms (above). We
will use only sinusoidal waves for simplicity.
9.2 Wave Equation
source: http://www.a-
levelphysicstutor.com/images/wa
ves/sinus-graph01.jpg
The left shows the
displacement-time
graph (1 particle, above)
and displacement-
distance graph (whole
wave, below)
Period is time taken for a point on the wave to complete one oscillation
Frequency is the no. of oscillations per unit time made by a point on the wave.
Wavelength is the distance between 2 adjacent points that are in phase.
Displacement of a particular point is the distance and direction of that point from its equilibrium position.
Amplitude is the maximum displacement of a point on the wave.
Crests are points with maximum, positive displacement.
Troughs are points with maximum, negative displacement.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 25 of 34
For a periodic wave, it travels one wavelength
during 1 period.
Hence, we can determine its speed:
Waves can be categorized into the following:
Mode of Vibration Longitudinal
Transverse
Motion Progressive
Stationary / Standing
Medium Mechanical
Electromagnetic
Matter
It is important to note that waves usually transfer
energy and not matter. Some waves require a
medium (e.g. sound and air/water) whereas
others can occur in vacuum (e.g. light rays from
Sun).
9.3 Transverse vs. Longitudinal Waves
9.3.1 Transverse Waves
Source:http://sciencecity.oupchina.com.hk/npaw/student/suppleme
ntary/images/graph-1b_8.jpg
Transverse waves are waves where displacement of particles is perpendicular to the direction of wave propagation.
Transverse waves are similar to their wave
profiles and they can (obviously) exist in many
planes. Examples include all electromagnetic
waves.
9.3.2 Longitudinal Waves
http://sciencecity.oupchina.com.hk/npaw/student/supplementary/i
mages/graph-1b_7.jpg
Longitudinal waves are waves where the displacement of particles is parallel to the direction of wave propagation.
Sound is a good example of longitudinal waves:
Source:http://hyperphysics.phy-
astr.gsu.edu/hbase/sound/imgsou/lwav2.gif
9.3.3 Progressive Waves
The displacement-distance graph shows the same
wave travelling left to right when t=0, t=1 and t=2.
Hence, by analyzing the particle at t= , we can
plot the displacement-time graph for the single
particle:
The same technique can be applied for
longitudinal waves by analyzing its wave profile.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 26 of 34
9.4 Phase Difference
The phase of an osicillation is the stage of oscillation that is represented by the phase angle, where 2 radians or 360 represents one complete cycle.
Note: points chosen must be in phase (black):
(
) (
)
9.5 Electromagnetic Waves
Source:http://micro.magnet.fsu.edu/primer/java/wavebasics/basicw
avesjavafigure1.jpg
Electromagnetic (EM) waves consist of the electric (E) and magnetic (B) field oscillating perpendicular to the direction of wave propagation. No medium is required. They travel at the speed of light (which is an EM
wave), where .
The EM spectrum classifies various EM waves:
Source: http://amazing-
space.stsci.edu/resources/qa/graphics/qa_emchart.gif
This table contains some uses of EM waves:
Name Detection Uses
Radio Radio Aerials Communications
Micro Tuned Cavities Communications and cooking
Infra-Red (IR)
Photography / Heating Effect
Satellite, TV controls
Visible Light
Eye / Photography
Sight, communication
Ultra Violet (UV)
Fluorescence, solid state detectors
Food sterilization
X-rays Fluorescence Diagnosis
Gamma ( rays
Scintillation counter
Radiotherapy
9.6 Intensity of Waves
Source:http://toonz.ca/bose/wiki/images/1/1e/IntensitySurfaceSphe
re.gif
The intensity is defined as the power per unit area that passes perpendicularly through a surface area, i.e.:
Since intensity is the energy per unit time per unit
area, we can thus conclude that:
Also, the diagram shows that we can apply the
inverse square law for intensity:
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 27 of 34
9.7 Polarization
Source:http://www.exo.net/~pauld/summer_institute/summer_day
8polarization/polarizerfencemodel600.jpeg
Polarization is the phenomenon where a transverse wave is made to oscillate in a single plane, the plane of polarization.
The first polarizer is known as the ‘polarizer’ and
the second is known as the ‘analyzer ’.
All polarization filters
only allow planes in the
plane of polarization to
pass. Hence, we can
resolve the electric field
to give a vertical and
horizontal component.
(
)
(
)
Hence, we observe the following:
If the polarizer and analyzer have planes of
polarization perpendicular to each other, then no
light passes through.
Chapter 10: Superposition
10.1 Principle of Superposition
The principle of superposition states that when 2 or more waves of the same kind overlap, the resultant displacement at any point any instant is given by the vector sum of individual displacements that each individual wave would cause at that instant, i.e.:
10.2 Interference
Inteference is the combination of waves in the same region of space at the same time to produce a resultant wave.
10.2.1 Constructive, Destructive Interference
There are 2 types of interference, constructive
and destructive interference:
10.2.2 Path Difference and Phase Difference
Source:http://roncalliphysics.wikispaces.com/file/view/nodal_lines.g
if/233899502/nodal_lines.gif
Plotting lines that join constructive interference
(red, anti-nodal lines) and destructive
interference (blue, nodal lines), we obtain the
above diagram.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 28 of 34
Path difference is defined as follows:
| | | |
For instance, the above diagram shows that:
| |
If they are in phase, constructive interference
occurs (as above). However, if they are anti-phase,
then destructive interference occurs.
Generalizing, we have the “final” phase
difference, given by:
Hence, for constructive interference,
Hence, for constructive interference,
(
)
10.3 Diffraction and Huygen’s-Fresnel
Principle
Diffraction is the apparent bending of waves around small obstacles and the spreading out of waves past small openings.
And we use the Huygen’s-Fresnel principle to
explain that phenomenon:
Huygen’s-Fresnel principle states that every point of a wave may be considered a secondary source of wavelets spreading out in all directions with a speed equal to the speed of propagation of the wave.
The following shows how it can apply to waves,
for both refraction and diffraction.
The new wave front is thus the envelope of
wavelets (green). Hence, for smaller apertures,
the envelope of wavelets is more spherical.
10.4 Young’s Double Slit Experiment
Thomas Young used the double-slit experiment in
1803 to show that light was a wave by
demonstrating intereference patterns predictable
by wave theory after his paper was rejected in
1799 by the royal society.
The single slit ensures the coherency of the wave.
2 waves are coherent if they have a constant (not necessarily 0) phase difference between them.
Coherent waves have the same wavelength and
frequency, and hence the same speed.
In order to determine maxima and minima, we
must first observe that for this experiment:
P
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 29 of 34
Source:http://www.u.arizona.edu/~mas13/draft4.310_files/image03
4.jpg (left)
First, we make the following observation:
For bright fringes (maxima),
For dark fringes (minima),
(
)
To find the exact positions of dark and bright
fringes (as shown in the initial experimental setup
diagram):
(
)
(
) (
)
Hence, the fringe separation, distance between 2
adjacent bright or dark fringes, is given by:
( )(
) (
)
10.5 Diffraction Grating
After seeing the double slit experiment, we now
use diffraction grating, adding many more
parallel, closely spaced and equidistance slits.
Diffraction grating usually involves hundreds or
thousands of slits.
Source:http://nothingnerdy.wikispaces.com/file/view/diffraction_gra
ting_geometry.jpg/213547792/diffraction_grating_geometry.jpg
Source: http://www.a-levelphysicstutor.com/wav-light-diffr.php
The above diagram represents the various
interference patterns with varying slits.
We observe that:
1) Maxima increases (more slits)
2) Better contrast in fringe pattern
3) Position of maxima is the same
Hence, using property 3, we can adapt the
equation for Young’s experiment to find the nth
order maxima for a diffraction grating, i.e.:
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 30 of 34
Source:http://hyperphysics.phy-
astr.gsu.edu/hbase/phyopt/imgpho/diffgrat.gif
It is important to observe that different
wavelengths of light have different maximas (e.g.
that of red and blue.
This is demonstrates that small angle
approximation does not hold (due to increasing
angle). Since fringes are irregularly spaced,
(
)
Commercially, gratings are labelled by no. of lines
per unit length, N.
10.6 Stationary Waves
Source:http://tap.iop.org/vibration/superpostion/324/img_full_4680
0.gif
Firstly, we note a phenomenon that when a wave
hits a fixed / denser surface (e.g. mirrors), they
undergo a phase change of radians.
This is because the wave exerts an upward force
(above diagram) on the fixed surface. Hence, by
Newton’s 3rd law, the wall exerts an equal and
opposite (downward) force on the medium (e.g.
string), resulting in a negative displacement.
Hence, when we have 2 identical waves moving in
opposite direction, we have a stationary wave
(red) that is being formed. Stationary waves
obviously have no translation of energy. It has the
following properties compared to a normal wave:
Nodes are points that never move and antinodes
are points having the greatest amplitude of
vibration.
10.7 Stretched Strings
Source:http://learn.uci.edu/media/OC08/11004/OC0811004_Standi
ngWave04.jpg
A string that is fixed on 2 ends can vibrate (above).
Like in simple harmonic motion, when the string
vibrates at its natural frequency, it obtains the
resonant modes of vibration (below):
Source:http://www.miqel.com/images_1/jazz_music_heart/harmoni
cs.jpg
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 31 of 34
Harmonics are all resonant frequencies of vibrations that can be generated.
( )
√
Note: the 1st harmonic is known as the
fundamental frequency.
Overtones are frequencies that can be produced by an instrument accompanying the 1st harmonic that is played.
10.8 Air Columns
There are 2 types of air columns:
10.8.1 Open pipes
Source:http://labspace.open.ac.uk/file.php/7027/ta212_2_015i.smal
l.jpg
Generalizing, we can deduce that:
10.8.2 Closed Pipes
Generalizing, we can deduce that:
10.8.3 End Corrections
End corrections occur because in practice, the
open end of a pipe is set into vibration and the
displacement antinode occurs at a distance c
(above).
From the above,
It has been found that end correction is
approximately 50-60% of the radius of the cross
sectional area of pipe. It might be better to take
them into consideration for large pipes.
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 32 of 34
Chapter 11: Miscellaneous
11.1 Useful Knowledge / Summary *
11.1.1 List of Useful Formulas by Topic
This list is non-exhaustive:
Physical Quantities and Measurements
| |
√
Kinematics
Dynamics
∑
∑
Forces
∑ ∑
Work, Energy, Power
Circular Motion
Gravitation
(
)
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 33 of 34
Gravitation (Continued)
∑( ∑
)
√ √
Oscillations / Simple Harmonic Motion
√
Waves
(
) (
)
Superposition
∑
| |
√
11.1.2 List of Useful Constants
These are fundamental constants to be used:
Gravitation constant
Speed of EM Waves
Electron Charge Planck’s Constant Stefan-Boltzmann Constant
Gas Constant Avogadro’s Constant
Boltzmann’s constant
H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved
Page 34 of 34
11.1.3 List of any other useful data (for now)
Credits
This set of physics notes is done by Ang Ray Yan,
Hwa Chong Institution 11S7B.
The following people deserve their due
recognition in making this set of notes:
- Mr Thomas, my physics tutor who rekindled
my interest for physics, showing me that
physics was useful, interesting, applicable and
unlike anything in my high school years.
- Lim Yao Chong for being a reliable helpline in
my weakest topics, particularly dynamics.
- Phang Zheng Xun for giving more accurate
definitions and various explanations.
- Yuan Yu Chuan for correcting my English,
which is of “powder-ful” standard.