Download Section 2. Mechanics Course Notes

A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
SECTION II
Newtonian Mechanics
CIE A-Level [AS and A2]
________________________
Course Notes
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
Syllabus Details______________________
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
3. Kinematics [AS]___________________________
Content
3.1 Linear motion
3.2 Non-linear motion
Learning outcomes_____________________________________
Candidates should be able to:
(a) define displacement, speed, velocity and acceleration
Symbol
Displacement
Velocity
Speed
Acceleration
Definition
Distance moved in particular direction
Velocity = change in displacement / time
Speed = total distance / time
Acceleration = change in velocity / time
s
v or u
v or u
a
SI
unit
m
ms-1
ms-1
ms-2
Vector /
Scalar
Vector
Vector
Scalar
Vector
(b) use graphical methods to represent displacement, speed, velocity and
acceleration
Displacement-Time Graphs
Stationary
Displacement / m
Constant +ve velocity
Constant -ve velocity
T ime / s
• G radient of a displacement-time graph = velocity
Velocity-Time Graphs
Acceleration-Time Graphs
Constant de-acceleration
T ime / s
• G radient of a velocity-time graph = acceleration
• Area under a velocity-time graph = displacement
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Acceleration / ms-2
Constant velocity
Velocity / ms-1
Constant acceleration
Acceleration
constantly increasing
Constant
acceleration
Acceleration constantly
decreasing
Time / s
•Area under an acceleration-time graph = change in velocity
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(c) find displacement from the area under a velocity-time graph
Velocity / ms-1
Velocity-Time Graphs
Time / s
Area under graph = displacement
(d) use the slope of a displacement-time graph to find the velocity
(e) use the slope of a velocity-time graph to find the acceleration
Velocity-Time Graphs
Velocity / ms-1
Displacement / m
Displacement-Time Graphs
s
t
T ime / s
Gradient = s/t = velocity
v
t
T ime / s
Gradient = v/t = acceleration
SEE PHET SIM
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(f) derive, from the definitions of velocity and acceleration, equations that
represent uniformly accelerated motion in a straight line
v=u+at
s=[(u+v)/2]t
v2=u2+2as
s=ut+1/2at2
s=vt-1/2at2
u
v
a
t
s
Initial velocity
Final velocity
Acceleration
Time
Displacement
DERIVATION…
Start with definition of acceleration
a = (v-u) / t
Rearrange to get first equation
v = u + at
Take definition of average velocity…
Average velocity = s / t
Average velocity = (v +u)/2
Therefore…
s / t = (v +u)/2
Rearrangement gives…
s = [(v + u)t]/2
Taking v = u + at and s = [(v + u)t]/2…. To eliminate v…
s = ut + 1/2at2
Taking v = u + at and s = [(v – u)t]/2…. To eliminate t…
v2=u2+2as
These formula are given on the test paper
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(g) solve problems using equations that represent uniformly accelerated motion in a
straight line, including the motion of bodies falling in a uniform gravitational field
without air resistance
If a body falls in a vacuum near the Earths surface it has an acceleration g of freefall
Displacement / m
IN A VACUUM
Velocity / ms-1
T ime / s
Acceleration / ms-2
T ime / s
g = ~10 ms-2
T ime / s
(h) recall that the weight of a body is equal to the product of its mass and the
acceleration of free fall
Mass = related to the amount of matter in an object
Weight = force of gravity exerted on an object (or the force on a supporting scale)
Weight (N) = mass (kg) x g (ms-2)
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g=acceleration of free fall
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(i) describe an experiment to determine the acceleration of free fall using a falling
body
Light Gates
Light gates record the time taken
for an object to pass.
Light
source
Light
detector
Strobe Photography
Strobe photography records
images at regular time intervals
t1
Ticker Tape
Ticker tapes have dots at regular
time intervals. The distance
between the dots can be
measured
1
t2
Falling object
s
t3
Velocity =
2
length of object
time through beam
If at 50Hz
5 spaces = 5/50 = 0.1 s
t4
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(j) describe qualitatively the motion of bodies falling in a uniform gravitational field
with air resistance
Displacement / m
WITH AIR RESISTANCE
Straight line as velocity
become constant
Velocity / ms-1
T ime / s
Ter minal Velocity
Acceleration / ms-2
T ime / s
Acceleration zero at
terminal velocity
T ime / s
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(k) describe and explain motion due to a uniform velocity in one direction and a
uniform acceleration in a perpendicular direction.
Projectile Motion
Gravitational field in vertical direction
vV
parabolic motion
vH
vH
vV
vV
vH
vV
vH
vV
vH
vHhas no force acting so is constant
vH
vV
vVhas a constant force acting so there is a constant acceleration


Parabolic in absence of air resistance
The vertical and horizontal components are independent
SEE PHET SIM
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
4. Dynamics [AS]____________________________
Content
4.1 Newton’s laws of motion
4.2 Linear momentum and its conservation
Learning outcomes_____________________________________
Candidates should be able to:
(a) state each of Newton’s laws of motion
NEWTON’S FIRST LAW:
“An object continues in uniform motion in a straight line or at rest unless a
resultant force acts.”
NEWTON’S SECOND LAW:
“The rate of change of momentum of an object is proportional to the resultant
force which acts on the object.”
NEWTON’S THIRD LAW:
“when two bodies A and B interact, the force that A exerts on B is equal and
opposite to the force B exerts on A.”
(b) show an understanding that mass is the property of a body that resists change in
motion
Mass is a property of a body that resists change in motion
a = 1m/s2
5N
10N
5kg
As the mass increases…
a = 1m/s2
• More force is needed for the same
acceleration
• The mass “resists” change in motion
10kg
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(c) describe and use the concept of weight as the effect of a gravitational field on a
mass
Gravitational field
No Gravitational Field
mass
mass
No Weight
Weight
Earth
(LARGE MASS)
Weight = force of gravity exerted on an object (or the force on a supporting scale)
Weight (N) = mass (kg) x g (N/kg)
g=gravitational field strength
(d) define linear momentum as the product of mass and velocity
Momentum = Mass x velocity
p (kgms-1) = m (kg) x v (ms-1)
(e) define force as rate of change of momentum
Force = change in momentum / time
F=p/t
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(f) recall and solve problems using the relationship F = ma, appreciating that
acceleration and force are always in the same direction
Example 1 – No friction
15N
F=ma
15N = 4kg x a
a = 15/4 = 3.75ms-2
4kg
Example 2 – With Friction
15N
4kg
Friction = 3N
F=ma
Resultant F = 15-3 = 12N
12N = 4kg x a
a = 12/4 = 3ms-2
Example 3 – On a slope
Normal reaction
F=ma
Force down the slope = 40(sin35 o)N = 22.9N
Resultant F = 22.9 - 3 = 19.9N
19.9N = 4kg x a
a (down slope) = 19.9/4 = 5.0ms-2
35 o
40N (mg = 4kg x 10ms
-2
)
REMEMBER: Acceleration is always in the direction of the resultant force
SEE PHET SIM
(g) state the principle of conservation of momentum
Law of conservation of linear momentum:
“The total linear momentum of a system of interacting particles remains
constant provided there is no resultant external force.”
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(h) apply the principle of conservation of momentum to solve simple problems
including elastic and inelastic interactions between two bodies in one dimension
(knowledge of the concept of coefficient of restitution is not required)
EXAMPLE…..
0ms-1
5ms-1
Total initial momentum = (mu)4kg + (mu)6kg
= 4x5 + 6x0 = 20 kgms-1
BEFORE
5ms-1
COLLISION
Total final momentum = (mv)10kg = 10 x v
?ms-1
Total initial momentum = Total final momentum
AFTER
20 = 10 x v
v = 20/10 = 2ms-1
(i) recognise that, for a perfectly elastic collision, the relative speed of approach is
equal to the relative speed of separation
u1
u2
v1
v2
Elastic Collisions
BEFORE
Collision
AFTER
Collision
Total initial momentum = Total final momentum
Total initial Kinetic Energy = Total final momentum Kinetic Energy
½ m1u12 + ½ m2u22 = ½ m1v12 + ½ m2v22
u1 – u2 = -(v1 – v2)
relative speed of approach = relative speed of separation
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(j) show an understanding that, while momentum of a system is always conserved in
interactions between bodies, some change in kinetic energy usually takes place.
ELASTIC AND INELASTIC Collisions…..
Initial Situation
v1
Elastic Collision
v1
No KE energy is lost
Inelastic Collision
v2
v3
Some KE lost
Total Momentum is conserved in all cases
mAv1 + mBv2 = mAv3 + mBv4
Inelastic Collisions are the normal situation
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
5. Forces [AS]_______________________________
Content
5.1 Types of force
5.2 Equilibrium of forces
5.3 Centre of gravity
5.4 Turning effects of forces
Learning outcomes_____________________________________
Candidates should be able to:
(a) describe the forces on mass and charge in uniform gravitational and electric
fields, as appropriate
Particle
Field
Effect
Uncharged mass
Gravitational
Uncharged mass
Charged mass
Electric field
Gravitational field
Positive charge
Electric field
Negative charge
Electric field
Attracted in direction of
field line
No effect
Attracted in direction of
field line
Attracted in direction of
field line
Repelled in opposite
direction to field line
(b) show an understanding of the origin of the upthrust acting on a body in a fluid
Upthrust
In a fluid...



Particles are constantly colliding with the sides of the container or
immersed object
These collisions produce a force
This force provides the upthrust on the immersed body
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(c) show a qualitative understanding of frictional forces and viscous forces including
air resistance (no treatment of the coefficients of friction and viscosity is required)



Frictional forces are forces that act against the direction of motion
Viscous forces result from motion through fluids
Both types of forces are due to the interaction between charges on the
moving object and the material it is close to
SEE PHET SIM
(d) use a vector triangle to represent forces in equilibrium
Forces in equilibrium
Situation
Force vectors
Vector triangle
Normal
reaction
Gravity
(e) show an understanding that the weight of a body may be taken as
acting at a single point known as its centre of gravity
Centre of Gravity
Centre of gravity
Each particle in a solid
has an associated weight
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The result of these forces
appears to act from a
single point
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(f) show an understanding that a couple is a pair of forces that tends to
produce rotation only
(g) define and apply the moment of a force and the torque of a couple
Moments, Couples, and Torque
Moment:
F2
distance
=Force x perpendicular distance
from pivot
Pivot
F2
distance
F1 = F2
Couple: a pair of equal but opposite
forces which gives a turning effect but no
resultant force
F1
Torque:
Moment of a couple = one force
x
perpendicular distance
between the forces
(h) show an understanding that, when there is no resultant force and no
resultant torque, a system is in equilibrium
Translational Equilibrium
If the resultant force on an object is
zero it said to be in translational
equilibrium

T, tension
F = zero
P, pull
W, weight
The ball is in
TRANS LATION
EQUILIBRIUM if…
Tsin = P
Tcos = W
(i) apply the principle of moments.
If an object is in equilibrium, the sum of the clockwise moments about a pivot are
equal to the sum of the anticlockwise moments.
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
6. Work, energy, power [AS]__________________
Content
6.1 Energy conversion and conservation
6.2 Work
6.3 Potential energy, kinetic energy and internal
energy
6.4 Power
Learning outcomes_____________________________________
Candidates should be able to:
(a) give examples of energy in different forms, its conversion and conservation, and
apply the principle of energy conservation to simple examples
Energy Type
Example
Kinetic Energy
Gravitational Potential Energy
Chemical Energy
Strain Energy
Nuclear Energy
Internal Energy
Electrical Energy
Light Energy
Sound Energy
Moving objects (Car)
Raised objects (Water in a dam)
Energy stored in bonds (coal, oil)
Energy due to flexing of materials (elastic band)
Energy associated with atomic nuclei (Fission reactors)
Energy of materials – kinetic from particles moving +
potential from bonds
Energy from moving charges (electricity)
Energy from Electromagnetic waves (light, IR)
Energy due to vibrating particles (sound)
Solar Energy
Photovoltaic Cell
Electrical Energy
Motor
Kinetic Energy
Potential Energy
Principles of Conservation of Energy:



Overall the total energy of any closed system must be constant
Energy is neither created or destroyed, it just changes form
There is no change in the total energy of the Universe
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(b) show an understanding of the concept of work in terms of the product of a force
and displacement in the direction of the force
Work = Force x distance moved by force
F
Work done = Fs cos 

s
(c) calculate the work done in a number of situations including the work done by a
gas that is expanding against a constant external pressure: W = p .V
W = F x
F
p=F/A
x
Constant pressure = p
F = pA
W = pAx
F
V = Ax
Work done = pV
This formula is given on the test paper
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(d) derive, from the equations of motion, the formula E k = ½ mv2
Kinetic Energy gained by an object is equal to the work done on that object
Start with…
v2 = u2 + 2as
Rearrange to form…
as = ½ u2 – ½ v2
Multiple both sides by mass…
mas = ½ mu2 – ½ mv2
F=ma so…
Fs = ½ mu2 – ½ mv2
(e) recall and apply the formula Ek = ½ mv2
Kinetic energy = energy associated with a moving object
Kinetic Energy (J) = 1/2mv2
m=mass, v=velocity
Kinetic Energy
v1
EK = 1/2mA v12
(f) distinguish between gravitational potential energy, electric potential energy and
elastic potential energy
Elastic Potential Energy
5 4 3
x
2 1
0
Gravitational Potential Energy
7 6
h
EP = mAgh
EElas = 1/2kx2
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
Electric Potential Energy
++++++++++++
q
EP = qV
----------------SEE PHET SIM
(g) show an understanding and use the relationship between force and potential
energy in a uniform field to solve problems
(h) derive, from the defining equation W = Fs, the formula Ep = mgh for potential
energy changes near the Earth’s surface
(i) recall and use the formula Ep = mgh for potential energy changes near the
Earth’s surface
W = Fs
Gravitational Potential Energy
Force needed to move mass mA is
equal to its weight
F = weight = mg
Mass mA moved through distance h
W = Fs = mgh
h
EP = mAgh
Work done = energy transfer
W = EP = mgh
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(j) show an understanding of the concept of internal energy
Internal Energy = Total Potential Energy + Total Random Kinetic Energy
Random Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic
Energy
Translational energy is the energy associated with the whole molecule moving
in a certain direction.
Rotational energy is the energy associated with the molecule rotation around a
certain point.
Potential energy is the energy associated with intermolecular forces.
(k) recall and understand that the efficiency of a system is the ratio of useful work
done by the system to the total energy input
Efficiency = Useful work OUT / Total energy transferred
Efficiency = Useful energy OUT / Total energy IN
Efficiency = Useful power OUT / Total power IN
(l) show an appreciation for the implications of energy losses in practical devices
and use the concept of efficiency to solve problems
Thermal Energy LOST
(Heat)
Kinetic Energy OUT
(Movement of car)
Chemical Energy IN
(Petrol)
Total energy in = total energy out
Chemical Energy = Kinetic Energy + Thermal Energy
Efficiency = Kinetic Energy Out / Chemical Energy In
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(m) define power as work done per unit time and derive power as the product of
force and velocity
Power = Rate at which Energy is transferred
Power (W) = energy transferred / time taken = work done / time taken
1 Watt (W) = 1 Js-1
Power = W/t
W = Fs
Power = Fs/t
Power = Fv
(n) solve problems using the relationships P = W/t and P = Fv.
SEE PAST PAPER QUESTION BOOKS
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
7. Motion in a circle
[A2]_____________________
Content
7.1 Kinematics of uniform circular motion
7.2 Centripetal acceleration
7.3 Centripetal force
Learning outcomes_____________________________________
Candidates should be able to:
(a) express angular displacement in radians
B

s
r
Angular displacement =  = s / r
 is measured in radians
A
(b) understand and use the concept of angular velocity to solve problems
Angular velocity = w =  / t
(c) recall and use v = rw to solve problems
v = wr
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(d) describe qualitatively motion in a curved path due to a perpendicular force, and
understand the centripetal acceleration in the case of uniform motion in a circle
 Force applied perpendicular to the velocity direction
 Acceleration in direction of force
Change in velocity directed in towards
the centre of the circle
vB
B
vB
vA
vA
A
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
Pendulum
car
ca
r

T
Tcos
Tsin
Centripetal force provided by
horizontal component of
tension
Centripetal force provided
by friction force between
tyres and road
mg
Car on a Corner
Solar System
Earth
Sun
Centripetal force provided by
gravitational attraction of Sun
SEE PHET SIM
(e) recall and use centripetal acceleration a = r w2, a = v2/r
Centripetal acceleration: The acceleration of an object travelling in circular motion.
Centripetal acceleration = acentripetal = v2/r
r = radius
acentripetal = r w2
(f) recall and use centripetal force F = mrw 2, F = mv2/2
Centripetal force = ma
= mv2/r
= m r w2
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
8. Gravitational field [A2]_____________________
Content
8.1 Gravitational field
8.2 Force between point masses
8.3 Field of a point mass
8.4 Field near to the surface of the Earth
8.5 Gravitational potential
Learning outcomes_____________________________________
Candidates should be able to:
(a) show an understanding of the concept of a gravitational field as an example of
field of force and define gravitational field strength as force per unit mass
g=F/m
g = gravitation field strength
m = test mass
units = N kg-1 = ms-2
(b) recall and use Newton’s law of gravitation in the form F = Gm1m2/r2
Newton’s law of universal gravitation: Every mass in the universe attracts all
the other masses in the universe.
Law of Universal Gravitation
m1
m2
force
force
r
F = Gm1 m 2
r
2
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F = Force
G = Universal gravitational constant
m = point mass
R = distance between point mass
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(c) derive, from Newton’s law of gravitation and the definition of gravitational field
strength, the equation g= GMr 2 for the gravitational field strength of a point mass
(d) recall and solve problems using the equation g= GM/r2 for the gravitational field
strength of a point mass
F = Gm1 m 2
r
2
=
+
g = Gm1
r
2
g=F/m
(e) show an appreciation that on the surface of the Earth g is approximately
constant and is called the acceleration of free fall
As changes in r at the surface of the Earth are small in comparison to the distance
to the center of the Earth (and so the center of gravity) g can be considered
constant.
g = 9.81 N/kg = 9.81 ms-1 = acceleration of free fall
(f) define potential at a point as the work done in bringing unit mass from infinity to
the point
(g) solve problems using the equation  = –GM/ r for the potential in the field of a
point mass
Gravitational potential energy: work done in moving a mass from infinity to a
point



Zero of potential energy is at infinity
Potential energy taken as a negative value
The work done in moving a mass between two points in a gravitational
field is independent of the path taken
Gravitational potential: Energy per unit test mass.
Gravitational potential: Energy per unit test mass.
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
potential energy decreases as
gravitational force does work
zero potential energy
at infinity
F1
F2
M
m
m
Force on mincreases
Gravitational potential energy of mass m=
 =
 =
-
GMm
r
work done
test mass
E
 = -
m
Gm
r
 = gravitational potential (Jkg-1)
m = mass producing field
This formula is given on the test paper
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(h) recognise the analogy between certain qualitative and quantitative aspects of
gravitational field and electric field
zero potential
at infinity
potential increase
F2
+Q
F1
q
q
Force on q increases
Electric Potential Energy of charge q =
Qq
r
Electric potential energy: work done in moving a charge from infinity to a point



Zero of potential energy is at infinity
Potential energy taken as a negative value
The work done in moving a charge between two points in an electril field
is independent of the path taken
Electric potential: Energy per unit test charge.
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
(i) analyse circular orbits in inverse square law fields by relating the gravitational
force to the centripetal acceleration it causes
Gravitational attraction = centripetal force
GMm
mv2
=
r
r2
GM = v2 r

v = GM
r
v = 2 r
T
GM = 2 r
T
2
( )
r3
= Constant
T2
 3
r =  r
T2
(j) show an understanding of geostationary orbits and their application.
Geostationary Orbit
Satellite
Earth’s
orbit
Satellite
orbit
Earths rotates once every 24hrs.
By using r3 / T2 the radius of orbit needed for geostationary orbit can be calculated
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A-Level Course Notes: PHYSICS
SECTION II: Newtonian Mechanics
Background Reading_________________
PHYSICS, Giancoli 6th edition, Chapter 2-8
Useful Websites______________________
http://phet.colorado.edu/en/simulations/category/new
http://www.s-cool.co.uk/alevel/physics.html
http://www.physicsclassroom.com/mmedia/index.cfm
http://www.phys.hawaii.edu/~teb/java/ntnujava/index.html
http://www.colorado.edu/physics/2000/index.pl
Constants___________________________
[These are given on each test paper]
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