1. Fundamentals of Physics: On-line lecture notes
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Fundamentals of Physics: On-line lecture notes
Prof. Paddy Regan
Dept. of Physics, University of Surrey, Guildford, GU2 7XH, UK

2. Fundamentals of Physics, Halliday, Resnick & Walker,
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Course textbook,
Fundamentals of Physics,
Halliday, Resnick & Walker,
published by Wiley & Sons. .

3. 4: Motion in 2 & 3 Dimensions
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1: Measurement
Units, length, time, mass
2: Motion in 1 Dimension
displacement, velocity, acceleration
3: Vectors
adding vectors & scalars, components, dot and cross products
4: Motion in 2 & 3 Dimensions
position, displacement, velocity, acceleration, projectiles, motion in a circle, relative motion
5: Force and Motion: Part 1
Newton’s laws, gravity, tension
6: Force and Motion: Part 2
Friction, drag and terminal speed, motion in a circle

4. 7: Kinetic Energy and Work
Work & kinetic energy, gravitational work, Hooke’s law, power.
8: Potential Energy and Conservation of Energy
Potential energy, paths, conservation of mechanical energy.
9: Systems of Particles
Centre of mass, Newton’s 2nd law, rockets, impulse,
10: Collisions.
Collisions in 1 and 2-D
11 : Rotation
angular displacement, velocity & acceleration, linear and angular relations, moment of inertia, torque.
12: Rolling, Torque and Angular Momentum
KE, Torque, ang. mom., Newton’s 2nd law, rigid body rotation
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5. 13: Equilibrium and Elasticity 14: Gravitation (and Electromagnetism)
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13: Equilibrium and Elasticity
equilibrium, centre of gravity, elasticity, stress and strain.
14: Gravitation (and Electromagnetism)
Newton’s law, gravitational potential energy, Kepler’s laws

6. REGAN PHY
1: Measurement
Physical quantities are measured in specific UNITS, i.e., by comparison to a reference STANDARD.
The definition of these standards should be practical for the measurements they are to describe (i.e., you can’t use a ruler to measure the radius of an atom!)
Most physical quantities are not independent of each other (e.g. speed = distance / time). Thus, it often possible to define all other quantities in terms of BASE STANDARDS including length (metre), mass (kg) and time (second).

7. SI Units Scientific Notation
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SI Units
The 14th General Conference of Weights and Measures (1971) chose
7 base quantities, to form the International System of Units
(Systeme Internationale = SI).
There are also DERIVED UNITS, defined in terms of BASE UNITS,
e.g. 1 Watt (W) = unit of Power = 1 Kg.m2/sec2 per sec = 1 Kg.m2/s3
Scientific Notation
In many areas of physics, the measurements correspond to very large
or small values of the base units (e.g. atomic radius ~ m).
This can be reduced in scientific notation to the ‘power of 10’ ( i.e.,
number of zeros before (+) or after (-) the decimal place).
e.g. 3,560,000,000m = x 109 m = 3.9 E+9m
& s = 4.92x10-7 s = 4.92 E-7s

8. Prefixes 1012 = Tera = T 109 = Giga = G 106 = Mega = M 103 = Kilo = k
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Prefixes
1012 = Tera = T
109 = Giga = G
106 = Mega = M
103 = Kilo = k
10-3 = milli = m
10-6 = micro = m
10-9 = nano = n
10-12 = pico = p
10-15 = femto = f
For convenience, sometimes,
when dealing with large or small
units, it is common to use a prefix
to describe a specific power of 10
with which to multiply the unit.
e.g.
1000 m = 103 m = 1E+3 m = 1 km
m = m = 0.1 nm

9. REGAN PHY
Converting Units
It is common to have to convert between different systems of units
(e.g., Miles per hour and metres per second). This can be done most
easily using the CHAIN LINK METHOD, where the original value
is multiplied by a CONVERSION FACTOR.
NB. When multiplying through using this method, make sure you
keep the ORIGINAL UNITS in the expression
e.g., 1 minute = 60 seconds, therefore (1 min / 60 secs) = 1
and (60 secs / 1 min) = 1
Note that 60 does not equal 1 though!
Therefore, to convert 180 seconds into minutes,
180 secs = (180 secs) x (1 min/ 60 secs) = 3 x 1 min = 3 min.

10. REGAN PHY
Length (Metres)
Original (1792) definition of a metre (meter in USA!) was
1/10,000,000 of the distance between the north pole and the equator.
Later the standards was changed to the distance between two lines
on a particular standard Platinum-Iridium bar kept in Paris.
(1960) 1 m redefined as 1,650, wavelengths of the (orange/red)
light emitted from atoms of the isotope 86Kr.
(1983) 1 m finally defined as the length travelled by light in vacuum
during a time interval of 1/299,792,458 of a second.
To Andromeda Galaxy ~ 1022 m
Radius of earth ~ 107 m
Adult human height ~ 2 m
Radius of proton ~ m

11. Time (Seconds) Standard definitions of the second ?
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Time (Seconds)
Standard definitions of the second ?
Original definition
1/(3600 x 24) of a day, 24 hours = 1day, 3600 sec per hours, thus
86,400 sec / day, 3651/4 days per year and 31,557,600 sec per year.
From HRW, p6
But, a day does not have
a constant duration!
(1967) Use atomic clocks,
to define 1 second as
the time for
9,192,631,770 oscillations
of the light of a specific
wavelength (colour) emitted
from an atom of caesium (133Cs)

12. Mass (Kg, AMU) Orders of Magnitude
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Mass (Kg, AMU)
1 kg defined by mass of Platinum-Iridium cylinder near to Paris.
Masses of atoms compared to each other for other standard.
Define 1 atomic mass unit = 1 u (also sometimes called 1 AMU) as
1/12 the mass of a neutral carbon-12 atom.
1 u = x kg
Orders of Magnitude
It is common for physicists to ESTIMATE the magnitude of
particular property, which is often expressed by rounding up (or
down) to the nearest power of 10, or ORDER OF MAGNITUDE,
e.g.. 140,000,000 m ~ 108m,

13. REGAN PHY
Estimate Example 1:
A ball of string is 10 cm in diameter, make an order of magnitude
estimate of the length, L , of the string in the ball. r d d

14. E.g., 2: Estimate Radius of Earth (from the beach!)
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E.g., 2: Estimate Radius of Earth (from the beach!) r d h q
q is the angle through which the
sun moves around the earth during the
time between the ‘two’ sunsets (t ~ 10 sec).

15. 2: Motion in a Straight Line
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2: Motion in a Straight Line
Position and Displacement.
To locate the position of an object we need to define this RELATIVE
to some fixed REFERENCE POINT, which is often called the
ORIGIN (x=0).
In the one dimensional case (i.e. a straight
line), the origin lies in the middle of an
AXIS (usually denoted as the ‘x’-axis)
which is marked in units of length.
Note that we can also define
NEGATIVE co-ordinates too.
x =
The DISPLACEMENT, Dx is the change from one position to
another, i.e., Dx= x2-x1 . Positive values of Dx represent motion in
the positive direction (increasing values of x, i.e. left to right looking
into the page), while negative values correspond to decreasing x.
Displacement is a VECTOR quantity. Both its size (or ‘magnitude’)
AND direction (i.e. whether positive or negative) are important.

16. Average Speed and Average Velocity
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Average Speed and Average Velocity
We can describe the position of an
object as it moves (i.e. as a function
of time) by plotting the x-position
of the object (Armadillo!) at different
time intervals on an (x , t) plot.
The average SPEED is simply the
total distance travelled (independent
of the direction or travel) divided by
the time taken. Note speed is a
SCALAR quantity, i.e., only its
magnitude is important (not its
direction).
From HRW

17. The average VELOCITY is defined by the displacement (Dx) divided
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The average VELOCITY is defined
by the displacement (Dx) divided
by the time taken for this
displacement to occur (Dt).
The SLOPE of the (x,t) plot gives average VELOCITY.
Like displacement, velocity is a
VECTOR with the same sign as
the displacement.
The INSTANTANEOUS VELOCITY
is the velocity at a specific
moment in time, calculated by making
Dt infinitely small (i.e., calculus!)

18. Acceleration HRW Acceleration is a change in
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Acceleration
Acceleration is a change in
velocity (Dv) in a given time (Dt).
The average acceleration, aav,
is given by
HRW
The instantaneous ACCELERATION
is given by a, where,
SI unit of acceleration is
metres per second squared (m/s2)

19. Constant Acceleration and the Equations of Motion
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Constant Acceleration and the Equations of Motion
For some types of motion (e.g., free
fall under gravity) the acceleration
is approximately constant, i.e., if
v0 is the velocity at time t=0, then
By making the assumption
that the acceleration is a
constant, we can derive a set
of equations in terms of the
following quantities
Usually in a given problem, three of these quantities are given
and from these, one can calculate the other two from the following
equations of motion.

20. Equations of Motion (for constant a).
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Equations of Motion (for constant a).

21. Alternative Derivations (by Calculus)
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Alternative Derivations (by Calculus)

22. Free-Fall Acceleration
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Free-Fall Acceleration
At the surface of the earth, neglecting any effect due to air resistance
on the velocity, all objects accelerate towards the centre of earth
with the same constant value of acceleration.
This is called FREE-FALL ACCELERATION, or ACCELERATION
DUE TO GRAVITY, g.
At the surface of the earth, the magnitude of g = 9.8 ms-2
Note that for free-fall, the equations of motion are in the y-direction
(i.e., up and down), rather than in the x direction (left to right).
Note that the acceleration due to gravity is always towards the centre
of the earth, i.e. in the negative direction, a= -g = -9.8 ms-2

23. REGAN PHY
Example
A man throws a ball upwards with an initial velocity of 12ms-1.
(a) how long does it take the ball to reach its maximum height ?
(b) what’s the ball’s maximum height ?
(a) since a= -g = -9.8ms-2,
initial position is y0=0 and
at the max. height vm a x=0
Therefore, time to max height from
(b)

24. ( c) How long does the ball take to reach a point 5m above its initial
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( c) How long does the ball take to reach a point 5m above its initial
release point ?
Note that there are TWO SOLUTIONS here (two different ‘roots’ to
the quadratic equation). This reflects that the ball passes the same
point on both the way up and again on the way back down.

25. REGAN PHY
3: Vectors
Quantities which can be fully described just by their size are called SCALARS.
Examples of scalars include temperature, speed, distance, time, mass, charge etc.
Scalar quantities can be combined using the standard rules of algebra.
A VECTOR quantity is one which need both a magnitude (size) and direction to be complete.
Examples of vectors displacement, velocity, acceleration, linear and angular momentum.
Vectors quantities can be combined using special rules for combining vectors.

26. Adding Vectors Geometrically
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Adding Vectors Geometrically
Any two vectors can be added using the
VECTOR EQUATION, where the sum of
vectors can be worked out using a triangle.
Note that two vectors can be
added together in either order to
get the same result. This is called
the COMMUTATIVE LAW.
Generally, if we have more than
2 vectors, the order of combination
does not affect the result. This is
called the ASSOCIATIVE LAW. =

27. Subtracting Vectors, Negative Vectors
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Subtracting Vectors, Negative Vectors
Note that as with all quantities, we can only add / subtract vectors
of the same kind (e.g., two velocities or two displacements).
We can not add differing quantities e.g., apples and oranges!)

28. Components of Vectors A simple way of adding vectors can
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Components of Vectors
A simple way of adding vectors can
be done using their COMPONENTS.
The component of a vector is the
projection of the vector onto the
x, y (and z in the 3-D case) axes in the
Cartesian co-ordinate system.
Obtaining the components is known as
RESOLVING the vector.
The components can be found using the rules
for a right-angle triangle. i.e. x y

29. REGAN PHY
Unit Vectors
A UNIT VECTOR is one whose magnitude is exactly equal to 1.
It specifies a DIRECTION. The unit vectors for the Cartesian
co-ordinates x,y and z are given by,
The use of unit vectors can make the
addition/subtraction of vectors simple.
One can simply add/subtract together
the x,y and z components to obtain
the size of the resultant component in
that specific direction. E.g, x y z 1

30. Vector Multiplication
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Vector Multiplication
There are TWO TYPES of vector multiplication.
One results in a SCALAR QUANTITY (the scalar or ‘dot’ product).
The other results in a VECTOR called the vector or ‘cross’ product.
For the SCALAR or DOT PRODUCT,

31. Vector (‘Cross’) Product
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Vector (‘Cross’) Product
The VECTOR PRODUCT of two vectors and
produces a third vector whose magnitude is given by
f is the angle between
the two initial vectors f
The direction of the
resultant is perpendicular
to the plane created by the
initial two vectors, such that

32. REGAN PHY
Example 1: y x q

33. Example 2:
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34. 4: Motion in 2 and 3 Dimensions
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4: Motion in 2 and 3 Dimensions
The use of vectors and their components is very useful for describing
motion of objects in both 2 and 3 dimensions.
Position and Displacement

35. Velocity and Acceleration
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Velocity and Acceleration
The average velocity is given by
While the instantaneous velocity is given by making Dt tend to 0, i.e.
Similarly, the average acceleration is given by,
While the instantaneous acceleration is given by

36. REGAN PHY
Projectile Motion
The specialist case where a projectile is ‘launched’ with an initial
velocity, and a constant free-fall acceleration, .
Examples of projectile motion are golf balls, baseballs, cannon balls.
(Note, aeroplanes, birds have extra acceleration see later).
We can use the equations of motion for constant acceleration and
what we have recently learned about vectors and their components
to analyse this type of motion in detail.
More generally,

37. Horizontal Motion Vertical Motion vy
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Horizontal Motion
In the projectile problem, there is NO ACCELERATION in the
horizontal direction (neglecting any effect due to air resistance).
Thus the velocity
component in the
x (horizontal) direction
remains constant throughout the flight, i.e.,
Vertical Motion vy

38. The Equation of Path for Projectile Motion
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The Equation of Path for Projectile Motion
Note that this is an
equation of the form
y=ax+bx2 i.e., a parabola
(also, often y0=x0=0.)

39. REGAN PHY
The Horizontal Range
Range
(y0,x0)
Max
height
vy=0

40. Example At what angle must a baseball be hit to make a home run if the
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At what angle must a baseball be hit to make a home run if the
fence is 150 m away ? Assume that the fence is at ground level, air resistance is negligible and the initial velocity of the baseball is 50 m/s. R
How far must the fence be moved back for no homers to be possible ?

41. Uniform Circular Motion
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Uniform Circular Motion
A particle undergoes UNIFORM CIRCULAR
MOTION is it travels around in a circular arc at a
CONSTANT SPEED. Note that although the
speed does not change, the particle is in fact
ACCELERATING since the DIRECTION OF THE
VELOCITY IS CHANGING with time.
The velocity vector is tangential to the instantaneous
direction of motion of the particle.
The (centripetal) acceleration is directed towards the centre of the circle
Radial vector (r) and the velocity vector (v) are always perpendicular

42. Proof for Uniform Circular Motion
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43. Relative Motion If we want to make measurements
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Relative Motion
If we want to make measurements
of velocity, position, acceleration etc.
these must all be defined RELATIVE to
a specific origin. Often in physical
situations, the motion can be broken down
into two frames of reference, depending
on who is the OBSERVER.
( someone who tosses a ball up in a
moving car will see a different motion
to someone from the pavement). A B p
If we assume that different FRAMES OF REFERENCE always move
at a constant velocity relative to each other, then using vector addition,
i.e., acceleration
is the SAME for
both frames of
reference!
(if VAB=const)!

44. 5: Force and Motion (Part 1)
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5: Force and Motion (Part 1)
If either the magnitude or direction of a particle’s velocity changes
(i.e. it ACCELERATES), there must have been some form of
interaction between this body and it surroundings. Any interaction
which causes an acceleration (or deceleration) is called a FORCE.
The description of how such forces act on bodies can be described by
Newtonian Mechanics first devised by Sir Isaac Newton ( )..
Note that Newtonian mechanics breaks down for (1) very fast speeds,
i.e. those greater than about 1/10 the speed of light c, c=3x108ms-1
where it is replaced by Einstein’s theory of RELATIVITY and (b)
if the scale of the particles is very small (~size of atoms~10-10m),
where QUANTUM MECHANICS is used instead.
Newton’s Laws are limiting cases for both quantum mechanics and
relativity, which are applicable for specific velocity and size regimes

45. Newton’s First Law Newton’s 1st law states
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Newton’s First Law
Newton’s 1st law states
‘ If no force acts on a body, then the body’s velocity can not change,
i.e., the body can not accelerate’
This means that
(a) if a body is at rest, it will remain at rest unless acted
upon by an external force, it; and
(b) if a body is moving, it will continue to move at that velocity
and in the same direction unless acted upon by an external force.
So for example,
(1) A hockey puck pushed across a ‘frictionless’ rink will move
in a straight line at a constant velocity until it hits the side of the rink.
(2) A spaceship shot into space will continue to move in the direction
and speed unless acted upon by some (gravitational) force.

46. Force Mass: we can define the mass of a body as the characteristic
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Force
The units of force are defined by the acceleration which that force
will cause to a body of a given mass.
The unit of force is the NEWTON (N) and this is defined by the
force which will cause an acceleration of 1 m/s2 on a mass of 1 kg.
If two or more forces act on a body we can find their resultant value
by adding them as vectors. This is known as the principle of
SUPERPOSITION. This means that the more correct version of
Newton’s 1st law is
‘ If no NET force acts on a body, then the body’s velocity can not
change, i.e., the body can not accelerate’
Mass: we can define the mass of a body as the characteristic
which relates the applied force to the resulting acceleration.

47. Newton’s 2nd Law Newton’s 2nd law states that
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Newton’s 2nd Law
Newton’s 2nd law states that
‘ The net force on a body is equal to the product of the body’s mass
and the acceleration of the body’
We can write the 2nd law in the form of an equation:
As with other vector equations, we can make three equivalent equations
for the x,y and z components of the force. i.e.,
The acceleration component on each axis is caused ONLY by the force
components along that axis.

48. the forces balance out each other and the body is in EQUILIBRIUM.
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If the net force on a body equals zero and thus it has no acceleration,
the forces balance out each other and the body is in EQUILIBRIUM.
We can often describe multiple forces acting on the same body using
a FREE-BODY DIAGRAM, which shows all the forces on the body.
137o
47o f y x
43o
HRW
p79

49. The Gravitational Force
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The Gravitational Force
The gravitational force on a body is the pulling force directed towards a second body. In most cases, this second body refers to the earth (or occasionally another planet).
From Newton’s 2nd law, the force is related to the acceleration by
A body’s WEIGHT equals the magnitude of the gravitational force on the body, i.e, W = mg. This is equal to the size of the net
force to stop a body falling to freely as measured by someone at
ground level. Note also that the WEIGHT MUST BE MEASURED WHEN THE BODY IS NOT ACCELERATING RELATIVE TO THE GROUND and that WEIGHT DOES NOT EQUAL MASS.
Mass on moon and earth equal but weights not ge=9.8ms-2, gm=1.7ms-2

50. REGAN PHY
The Normal Force
The normal force is the effective ‘push’ a body feels from a body
to stop the downward acceleration due to gravity, for example the
upward force which the floor apparently outs on a body to keep it
stationary against gravity.
General equation for block on a table is
Note the NORMAL FORCE is ‘normal’
(i.e. perpendicular) to the surface.

51. REGAN PHY
Example
A person stands on a weighing scales in a lift (elevator!) What is the
general solution for the persons measured weight on the scales ?
So, if lift accelerates upwards (or the downward speed decreases!) the persons
weight INCREASES, if the lift accelerates
downwards (or decelerates upwards)
the persons weight DECREASES compared to the stationary (or constant velocity) situation.

52. REGAN PHY
Tension
Tension is the ‘pulling force’ associated with a rope/string pulling a
body in a specific direction. This assumes that the string/rope is taught
(and usually also massless).
For a frictionless surface and a massless, frictionless pulley, what are the accelerations of the sliding and hanging blocks and the tension in the cord ? M m

53. REGAN PHY
Newton’s Third Law
Two bodies interact when they push or pull on each other. This leads
to Newton’s third law which states,
‘ When two bodies interact, the forces on the bodies from each other
are always equal in magnitude and opposite in direction ’
The forces between two interacting
bodies are called a ‘third-law pair
forces’.
e.g., Table pushes up block with
force N, block pushes down table
with force Fg, where Fg=N
Sometimes this is differently stated as
‘ for every action there is an equal
but opposite reaction ’

54. Example mg N T y x 50o mg sin 40o T cos50o N T mg = g x 15kg 40 o
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Example mg N T y x
40 o
50o
mg sin 40o
T cos50o N T
mg =
g x 15kg
q=40 o

55. 6: Force and Motion (Part II)
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6: Force and Motion (Part II)
Friction: When two bodies are in contact, the resistance to movement
between their surfaces is known as FRICTION. The properties of
frictional forces are that if a force, F, pushes an object along a surface
(e.g., a block along a surface),
1) If the body does not move, the STATIC FRICTIONAL FORCE,
fs is equal in magnitude and opposite in direction to the
component of the pushing force, F, along the surface.
2) The magnitude of the frictional force, fs, has a maximum value,
f s,max, which is given by f s,max=msN where ms is the coefficient
of static friction.
3) If the body begins to move along the surface, the magnitude of the
frictional force reduces to fk=mkN, where mk is the coefficient of
kinetic friction.

56. Drag Force and Terminal Speed
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Drag Force and Terminal Speed
When a body passes through a fluid (i.e., gas or a liquid) such as a ball
falling through air, if there is a relative velocity between the body and
the fluid, the body experiences a DRAG FORCE which opposes this
relative motion and is in the opposite direction to the motion of the
body (i.e., in the direction which the fluid flows relative to the body).
The magnitude of this drag force is related
to the relative speed of the body in the fluid
by a DRAG COEFFICIENT, C, which is
experimentally determined. The magnitude
of the drag force is given by the expression for D, which depends
on the fluid density (i.e. mass per unit volume, r), the effective
cross-sectional area, A (i.e. the cross-sectional area perpendicular to the
direction of the velocity vector), and the relative speed, v.

57. a quantity associated with a body which can varies with the speed, v.
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Note that the drag coefficient. C, is not really a constant, but rather
a quantity associated with a body which can varies with the speed, v.
(for the purposes of this course, however, assume C = constant).
The direction of the drag force is opposed to the motion of the object
through the fluid. If a body falls through air, the drag force due to the
air resistance will start at zero (due to zero velocity) at the start of the
fall, increasing as the downward velocity of the falling body increases.

58. Forces in Uniform Circular Motion
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Forces in Uniform Circular Motion
Recalling that for a body moving in a circular arc or radius, r, with
constant speed, v, the MAGNITUDE of the ACCELERATION, a, is
given by a = v2/r, where a is called the centripetal acceleration.
We can say that a centripetal force accelerates a body by changing the
direction of that body’s velocity without changing its speed.
Note that this centripetal force is not a ‘new’ force, but rather a
consequence of another external force, such as friction, gravity or
tension in a string.
Examples of circular motion are
(1) Sliding across your seat when your car rounds a bend:
The centripetal force (which here is the frictional force between
the car wheels and the road) is enough to cause the car to accelerate
inwards in the arc. However, often the frictional force between you
and your seat is not strong enough to make the passenger go in this
arc too. Thus, the passenger slides to the edge of the car, when its push
(or normal force) is strong enough to make you go around the arc.

59. (2) the (apparent) weightlessness of astronauts on the space shuttle.
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(2) the (apparent) weightlessness of astronauts on the space shuttle.
Here the centripetal force (which causes the space shuttle to orbit the
earth in a circular orbits) is caused by the gravitational force of the
earth on all parts of the space shuttle (including the astronauts).The
centripetal force is equal on all areas of the astronauts body so he/she
feels no relative extra pull etc. on any specific area, giving rise to a
sensation of weightlessness.
Note that the magnitude of the centripetal FORCE is given, (from
Newton’s second law) by : F = ma = m v2/r
Note that since the speed, radius and mass are all CONSTANTS so
is the MAGNTIUDE OF THE CENTRIPETAL FORCE. However,
DIRECTION IS NOT CONSTANT, varying continuously so as to
point towards the centre of a circle.

60. Example: r At what constant speed does the roller
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Example: r
At what constant speed does the roller
coasters have to go to ‘loop the loop’
of radius r ?
At the top of the loop, the free body
forces on the roller coaster are gravity
(downwards) and the normal
force (also inwards). The total acceleration is
also inwards (i.e., in the downwards direction). N Fg

61. 7: Kinetic Energy and Work
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7: Kinetic Energy and Work
One way to describe the motion of objects is by the use of Newton’s
Laws and Forces. However, an alternative way is describe the motion
in terms of the ENERGY of the object.
The KINETIC ENERGY (K) is the energy associated with the
MOTION of an object. It is related to the mass and velocity of a body
by K= 1/2 mv2 , where m and v are the mass and velocity of the body.
The SI unit of energy is the Joule (J) where 1 Joule = 1kg.m2s-2.
Work:
`Work is the energy transferred to or from an object by means of a
force acting on it. Energy transferred to the object is positive work,
while energy transferred from the object is negative work.’
For example, if an object is accelerated such that it increases its
velocity, the force has ‘done work’ on the object.

62. Work and Kinetic Energy
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Work and Kinetic Energy
The work done (W) on an object by a force, F, causing a displacement,
d, is given by the SCALAR PRODUCT, W = F.d =dFcosf where
Fcosf is the component of the force along the object’s displacement.
This expression assumes a CONSTANT FORCE (one that does not
change in magnitude or direction) and that the object is RIGID (all
parts of the object move together).
Example: If an object moves in a straight line with initial velocity, v0
and is acted on by a force along a distance d during which the velocity
increases to v due to an acceleration, a, from Newton’s 2nd Law the
magnitude of the force is given by F = max . From the equations of
motion v2=vo2+2axd . By substituting for the acceleration, ax, we have,
is the Work-Kinetic Energy Theorem

63. Work Done by a Gravitational Force.
REGAN PHY
Work Done by a Gravitational Force.
If an object is moved upwards against gravity, work must be done.
Since the gravitational force acts DOWNWARDS, and equals Fgr=mg ,
the work done in moving the object upwards in the presence of this
force is W=F.d = mg . d where d is the (vector) displacement in the
upward direction, (which we assume is the positive y-axis).

64. Work Done Lifting and Lowering an Object.
REGAN PHY
Work Done Lifting and Lowering an Object.
If we lift an object by applying a vertical (pushing) force, F, during
the upward displacement, work (Wa) is done on the object by this
applied force. The APPLIED FORCE TRANSFERS ENERGY TO
the object, while the GRAVITY TRANSFERS ENERGY FROM it.

65. Spring Forces and Hooke’s Law
REGAN PHY
Spring Forces and Hooke’s Law
The spring force is an example of a VARIABLE FORCE.
For a PERFECT SPRING, stretching or compressing gives rise to
RESTORING FORCE which is proportional to the displacement
of the spring from its relaxed state. This is written by Hooke’s Law
(after Robert Hooke, 17th century British scientist) as
The work done by a perfect spring can not be obtained from F.d, as the
force is not constant with d. Instead, the work done over the course of
the extension/compression must be summed incrementally.

66. Work Done by an Applied Force
REGAN PHY
Work Done by an Applied Force
During the displacement of the spring, the applied force, Fa, does
work, Wa on the block and the spring restoring force, Fs does work Ws.
If the block attached to a spring is stationary before and after its
displacement, then the work done on the spring by the applied force
is the negative of the work done on it by the spring restoring force.

67. Work Done by a General Variable Force.
REGAN PHY

68. Work-Kinetic Energy Theorem with a General,Variable Force
REGAN PHY

69. Power The INSTANTANEOUS POWER is given by
REGAN PHY
Power
POWER is the RATE AT WHICH WORK IS DONE. The AVERAGE
POWER done due to a force responsible for doing work, W in a time
period, Dt is given by Pave = W/D t .
The INSTANTANEOUS POWER is given by
The SI unit of power = Watt (W), where 1 W= 1 J per sec=1 kg.m2/s3
Note that the imperial unit of horsepower (hp) is still used, for
example for cars. 1hp = 746 W
The amount of work done is sometimes expressed as the product of the
power output multiplied by time taken for this. A common unit for this
is the kilowatt-hour, where 1kWh = 1000x3600 J = 3.6 x106J = 3.6MJ.
We can also describe the instantaneous power in terms of rate at
which a force does work on a particle,

70. REGAN PHY
Example 1:
What is the total energy associated with a collision between two
locomotives, at opposite ends of a 6.4km track accelerating towards
each other with a constant acceleration of 0.26 m/s2 if the mass of
each train was 122 tonnes (1 tonne =103kg) ?

71. Example 2: If a block slide across a frictionless
REGAN PHY
Example 2:
If a block slide across a frictionless
floor through a displacement of
-3m in the direction, while at the same
time a steady (i.e. constant) force of
F=(2i-6j) Newtons pushes against the
crate,
(a) How much work does the wind force do
on the crate during this displacement ?
(b) If the crate had a kinetic energy of 10J at the start of the
displacement, how much kinetic energy did it have at the end of the
-3m ?

72. Example 3: If a block of mass, m, slides across a
REGAN PHY
Example 3:
If a block of mass, m, slides across a
frictionless floor with a constant speed
of v until it hits and compresses a perfect
spring, with a spring constant, k.
At the point where the spring is compressed
such that the block is momentarily stopped,
by what distance, x, is the spring compressed ? m k v x
v=0

73. 8: Potential Energy & Conservation of Energy
REGAN PHY
8: Potential Energy & Conservation of Energy
Potential energy (U) is the energy which can be associated with
configuration of a systems of objects.
One example is GRAVITATIONAL POTENTIAL ENERGY,
associated with the separation between two objects attracted to each
other by the gravitational force. By increasing the distance between
two objects (e.g. by lifting an object higher) the work done on the
gravitational force increases the gravitational potential energy of the
system.
Another example is ELASTIC POTENTIAL ENERGY which is
associated with compression or extension of an elastic object (such as
a perfect spring). By compressing or extending such a spring, work
is done against the restoring force which in turn increases the elastic
potential energy in the spring.

74. Work and Potential Energy
REGAN PHY
In general, the change in potential energy, DU is equal to the negative
of the work done (W) by the force on the object (e.g., gravitational
force on a falling object or the restoring force on a block pushed by a
perfect spring), i.e., DU=-W
Conservative and Non-Conservative Forces
If work, W1, is done, if the configuration by which the work is done is
reversed, the force reverses the energy transfer, doing work, W2.
If W1=-W2, whereby kinetic energy is always transferred to potential
energy, the force is said to be a CONSERVATIVE FORCE.
The net work done by a conservative force in a closed path is zero.
The work done by a conservative force on a particle moving between
2 points does not depend on the path taken by the particle.
NON-CONSERVATIVE FORCES include friction, which causes
transfer from kinetic to thermal energy. This can not be transferred
back (100%) to the original mechanical energy of the system.

75. Determining Potential Energy Values
REGAN PHY
Determining Potential Energy Values

76. Conservation of Mechanical Energy
REGAN PHY
Conservation of Mechanical Energy

77. The Potential Energy Curve
REGAN PHY
The Potential Energy Curve
In the general, the force at position x,
can be calculated by differentiating
the potential curve with respect to x
(remembering the -ve sign). F(x) is minus
the SLOPE of U(x) as a function of x

78. Turning Points For conservative forces, the
REGAN PHY
Turning Points
For conservative forces, the
mechanical energy of the system
is conserved and given by,
U(x) + K(x) = Emec
where U(x) is the potential energy
and K(x) is the kinetic energy.
Therefore, K(x) = Emec-U(x).
Since K(x) must be positive ( K=1/2mv2),
the max. value of x which the particle
has is at Emec=U(x) (i.e., when K(x)=0).
Note since F(x) = - ( dU(x)/dx ) ,
the force is negative.
Thus the particle is ‘pushed back.
i.e., it turns around at a boundary.

79. REGAN PHY
Equilibrium Points
Equilibrium Points: refer to points where, dU/dx=-F(x)=0.
Neutral Equilibrium: is when a particle’s total mechanical energy is
equal to its potential energy (i.e., kinetic energy equals zero). If no
force acts on the particle, then dU/dx=0 (i.e. U(x) is constant) and
the particle does not move. (For example, a marble on a flat table top.)
Unstable Equilibrium: is a point where the kinetic energy is
zero at precisely that point, but even a small displacement from this
point will result in the particle being pushed further away (e.g., a
ball at the very top of a hill or a marble on an upturned dish).
Stable Equilibrium: is when the kinetic energy is zero, but any
displacement results in a restoring force which pushes the particle
back towards the stable equilibrium point. An example would be a
marble at the bottom of a bowl, or a car at the bottom of a valley.

80. Particles at A,B, C and D are in at equilibrium points where dU/dx = 0
REGAN PHY A B C D x
U(x)
Particles at A,B, C and D are in at equilibrium points
where dU/dx = 0
A,C are both in stable equilibrium ( d 2U/dx2 = +ve )
B is an unstable equilibrium ( d 2U/dx2 = -ve )
D is a neutral equilibrium ( d 2U/dx2 = 0 )

81. Work Done by an External Force
REGAN PHY
Work Done by an External Force
Previously we have looked at the work done to/from an object.
We can extend this to a system of more than one object.
Work is the energy transferred to or from a system by means of an
external force acting on that system.
No friction (conservative forces)
Including friction

82. Conservation of Energy
REGAN PHY
Conservation of Energy
This states that
‘ The total energy of a system, E, can only change by amounts of
energy that are transferred to or from the system. ’
Work done can be considered as energy transfer, so we can write,
If a system is ISOLATED from it surroundings, no energy can be
transferred to or from it. Thus for an isolated system, the total energy
of the system can not change, i.e.,
Another way of writing this is,
which means that for an isolated system, the total energies can be
related at different instants, WITHOUT CONSIDERING THE
ENERGIES AT INTERMEDIATE TIMES.

83. Example 1: A child of mass m slides down a helter
REGAN PHY
Example 1:
A child of mass m slides down a helter
skelter of height, h. Assuming the
slide is frictionless, what is the speed of
the child at the bottom of the slide ?
h=10m

84. Example 2: A man of mass, m, jumps from a
REGAN PHY
Example 2:
A man of mass, m, jumps from a
ledge of height, h above the ground,
attached by a bungee cord of length
L. Assuming that the cord obeys
Hooke’s law and has a spring constant,
k, what is the general solution for the
maximum extension, x, of the cord ? L h x m

85. REGAN PHY
9: Systems of Particles
Centre of Mass (COM): The COM is the point that moves as though
all the mass of a body were concentrated there.

86. Centre of Mass for Solid Bodies
REGAN PHY
Centre of Mass for Solid Bodies

87. Newton’s 2nd Law for a System of Particles.
REGAN PHY
Newton’s 2nd Law for a System of Particles.

88. Linear Momentum Thus we can re-write Newton’s 2nd law as
REGAN PHY
Linear Momentum
Thus we can re-write Newton’s 2nd law as
‘ The rate of change of the linear momentum with respect to time is
equal to the net force acting on the particle and is in the direction of
the force.’
The linear momentum of a system of particles is equal to the product of
the total mass of the system, M, and the velocity of the centre of mass,

89. Conservation of Linear Momentum
REGAN PHY
Conservation of Linear Momentum
This is the law of CONSERVATION OF LINEAR MOMENTUM
which we can write in words as
‘In no net external force acts on a system of particles, the total linear
momentum, P , of the system can not change.’
also, leading on from this,
‘ If the component of the net external force on a system is zero along
a specific axis, the components of the linear momentum along that
axis can not change.’

90. Varying Mass: The Rocket Equation
REGAN PHY
Varying Mass: The Rocket Equation
For rockets, the mass of the rocket is is not constant, (the rocket fuel is
burnt as the rocket flies in space). For no gravitational/drag forces, M v
M+dm
v+dv
-dm U
a) time = t
b) time = t+dt
1st rocket
equation

91. REGAN PHY
2nd rocket
equation

92. Internal Energy Changes and External Forces
REGAN PHY
Internal Energy Changes and External Forces
Energy can be transferred ‘inside a system’ between internal and
mechanical energy via a force, F. (Note that up to now each part of an
object has been rigid). In this case, the energy is transferred internally,
from one part of the body to another by an external force.

93. 10: Collisions
REGAN PHY
‘A collision is an isolated event in which two or more colliding
bodies exert forces on each other for a short time.’
Impulse
-F(t)
F(t)

94. back in the opposite direction with the same magnitude of speed
REGAN PHY
E.g A 140g is pitched with a horizontal speed of vi=39m/s. If it is hit
back in the opposite direction with the same magnitude of speed
what is the impulse, J, which acts on the ball ?

95. Momentum and Kinetic Energy in Collisions
REGAN PHY
Momentum and Kinetic Energy in Collisions
In any collision, at least one of the bodies must be moving prior to
the collision, meaning that there must be some amount of kinetic
energy in the system prior to the collision. During the collision, the
kinetic energy and linear momentum are changed by the impulse from
the other colliding body.
If the total kinetic energy of the system is equal before and after
collision, it is said to be an ELASTIC COLLISION.
However, in most everyday cases, some of this kinetic energy is
transferred into another form of energy such as heat or sound.
Collisions where the kinetic energies are NOT CONSERVED
are known as INELASTIC COLLISIONS.
In a closed system, the total linear momentum, P of the system can
not change, even though the linear momentum of each of the
colliding bodies may change.

96. REGAN PHY

97. Elastic Collisions in 1-D
REGAN PHY
Elastic Collisions in 1-D
In an elastic collision, the total energy before the collision is equal to
the total kinetic energy after the collision. Note that the kinetic energy
of each body may change, but the total kinetic energy remains constant.
m1, v1,i
m2, v2,i=0
before elastic collision
m1, v1,f
m2, v2,f
after elastic collision

98. REGAN PHY

99. REGAN PHY
Example 1:
Nuclear reactors require that the energies of neutrons be reduced by
nuclear collisions with a MODERATOR MATERIAL to low energies
(where they are much more likely to take part in chain reactions). If the
mass of a neutron is 1u~1.66x10-27kg, what is the more efficient
moderator material, hydrogen (mass = 1u) or lead (mass~208u)?
Assume the neutron-moderator collision is head-on and elastic.

100. Example 2: The Ballistic Pendulum
REGAN PHY
Example 2: The Ballistic Pendulum
A ballastic pendulum uses the transfer
of energy to measure the speed of
bullets fired into a wooden block
suspended by string.
vbul
Mblock h

101. 1-D Collisions with a Moving Target
REGAN PHY
m1, v1,i
m2, v2,i
before elastic collision

102. Collisions in Two Dimensions
REGAN PHY
Collisions in Two Dimensions
When two bodies collide, the
impulses of each body on the other
determine the final directions following
the collision. If the collision is not
head-on (i.e. not the simplest 1-D case)
in a closed system, momentum remains
conserved, thus, for an elastic collision
where Ktot,I=Ktot,f , we can write,
m1, v1,i
m2, v2,i
m1, v1,f
m2, v2,f q1 q2 y x
For a 2-D glancing collision, the collision can be described in terms of
momentum components. For the limiting case where the body of m2 is
initially at rest, if the initial direction of mass, m1 is the x-axis, then,

103. REGAN PHY
11: Rotation
Most motion we have discussed thus far refers to translation.
Now we discuss the mechanics of ROTATION, describing
motion in a circle.
First, we must define the standard rotational properties.
A RIGID BODY refers to one where all the parts rotate
about a given axis without changing its shape.
(Note that in pure translation, each point moves the same
linear distance during a particular time interval).
A fixed axis, known as the AXIS OF ROTATION is
defined by one that does not change position under rotation.
Each point on the body moves in a circular path described by an
angular displacement Dq. The origin of this circular path is centred at
the axis of rotation.

104. Summary of Rotational Variables
REGAN PHY
Summary of Rotational Variables
All rotational variables are defined relative to motion about a
fixed axis of rotation.
The ANGULAR POSITION, q, of a body is then the angle between
a REFERENCE LINE, which is fixed in the body and perpendicular
to the rotation axis relative to a fixed direction (e.g., the x-axis).
If q is in radians, we know that q=s/r where s is the length of arc
swept out by a radius r moving through an angle q. (Note
counterclockwise represent increase in positive q).
axis of
rotation
reference
line q s r
Radians are defined by s/r and are thus
pure, dimensionless numbers without
units. The circumference of a circle
(i.e., a full arc) s=2pr, thus in radians,
the angle swept out by a single, full
revolution is 360o = 2pr/r=2p. Thus,
1 radian = 360 / 2p = 57.3o
= of a complete revolution. x

105. The angular displacement, Dq represents the change in
the angular position due to rotational motion.
In analogy with the translational motion variables, other angular
motion variables can be defined in terms of the change (Dq), rate of
change (w ) and rate of rate of change (a ) of the angular position.
REGAN PHY

106. Relating Linear and Angular Variables
REGAN PHY
For the rotation of a rigid body, all of the particles in the body take the
same time to complete one revolution, which means that they all have
the same angular velocity,w, i.e., they sweep out the same measure of
arc, dq in a given time. However, the distance travelled by each of
the particles, s, differs dramatically depending on the distance, r, from
the axis of rotation, with the particles with the furthest from the axis
of rotation having the greatest speed, v.
at and ar are the tangential and radial accelerations respectively.
We can relate the rotational and linear variables using the following
(NB.: RADIANS MUST BE USED FOR ANGULAR VARIABLES!)

107. Rotation with Constant Acceleration
REGAN PHY
Rotation with Constant Acceleration
For translational motion we have seen that for the case of a
constant acceleration, we can derive a series of equations of motion.
By analogy, for CONSTANT ANGULAR ACCELERATION, there
is a corresponding set of equations which can be derived by
substituting the translational variable with its rotational analogue.
TRANSLATIONAL
ROTATIONAL

108. Example 1: A grindstone rotates at a constant
REGAN PHY
axis of
rotation
ref. line
for q0=0
A grindstone rotates at a constant
angular acceleration of a=0.35rad/s2.
At time t=0 it has an angular velocity
of w0=-4.6rad/s and a reference line on
its horizontal at the angular position, q0=0.
(a) at what time after t=0 is the reference line at q=5 revs ?
Note that while w0 is negative, a is positive. Thus the grindstone starts
rotating in one direction, then slows with constant deceleration before
changing direction and accelerating in the positive direction.
At what time does the grindstone momentarily stop to reverse direction?

109. Kinetic Energy of Rotation
REGAN PHY

110. Calculating to the Rotational Moment of Inertia
REGAN PHY
Calculating to the Rotational Moment of Inertia
The Parallel-Axis Theorem
To calculate I if the moment of inertia about a parallel axis passing
through the body’s centre of mass is known, we can use I=Icom+Mh2,
where, M= the total mass of the body, h is the perpendicular distance
between the parallel centre of mass axis and the axis of rotation and
Icom is the moment of inertia about the centre of mass axis.

111. REGAN PHY
Example 2:
The HCl molecule consists of a hydrogen atom (mass 1u) and a
chlorine atom (mass 35u). The centres of the two atoms are separated
by 127pm (=1.27x10-10m). What is the moment of inertia, I, about an
axis perpendicular to the line joining the two atoms which passes
through the centre of mass of the HCl molecule ?
com d a
d-a
rotation axis Cl H

112. Torque and Newton’s 2nd Law
REGAN PHY
Torque and Newton’s 2nd Law
The ability of a force, F, to rotate an object
depends not just on the magnitude of its
tangential component, Ft but also on how
far the applied force is from the axis of
rotation, r. The product of Ft r =Frsinf
is called the TORQUE (latin for twist!) t . r F
Frad Ft f O

113. Work and Rotational Kinetic Energy
REGAN PHY

114. 12: Rolling, Torque and Angular Momentum
REGAN PHY
12: Rolling, Torque and Angular Momentum
Rolling: Rolling motion (such as
a bicycle wheel on the ground) is
a combination of translational
and rotational motion.
COM
motion. P O R q S

115. The kinetic energy of rolling. A rolling object has two types of
REGAN PHY
The kinetic energy of rolling.
A rolling object has two types of
kinetic energy, a rotational
kinetic energy due to the
rotation about the centre of
mass of the body and
translational kinetic energy due to the translation of its centre of mass.
COM
motion. P O R q S

116. If a wheel rolls at a constant speed, it
Rolling Down a Ramp
REGAN PHY q R P
If a wheel rolls at a constant speed, it
has no tendency to slide. However, if this
wheel is acted upon by a net force (such
as gravity) this has the effect of speeding
up (or slowing down) the rotation, causing
an acceleration of the centre of mass of the system, acom along the
direction of travel. It also causes the wheel to rotate faster. These
accelerations tend to make the wheel SLIDE at the point, P, that it
touches the ground. If the wheel does not slide, it is because the
FRICTIONAL FORCE between the wheel and the slide opposes the
motion. Note that if the wheel does not slide, the force is the STATIC
FRICTIONAL FORCE ( fs ).

117. Rolling down a ramp (cont.)
REGAN PHY
Rolling down a ramp (cont.)
For a uniform body of mass, M and radius, R,
rolling smoothly (i.e. not sliding) down a ramp
tilted at angle, q (which we define as the x-axis
in this problem), the translational acceleration down the ramp can be calculated, from q R P

118. REGAN PHY R0 R w T Mg

119. Example 1: A uniform ball of mass M=6 kg and radius R rolls
REGAN PHY
Example 1:
A uniform ball of mass M=6 kg and radius R rolls
smoothly from rest down a ramp inclined at
30o to the horizontal.
(a) If the ball descends a vertical height of
1.2m to reach the bottom of the ramp, what is the
speed of the ball at the bottom ?
1.2m

120. Example 1 (cont):
REGAN PHY
(b) A uniform ball, hoop and disk, all of mass M=6 kg and
radius R roll smoothly from rest down a ramp inclined
at 30o to the horizontal. Which of the three objects reaches
the bottom of the slope first ?
1.2m

121. Torque was defined previously for a rotating rigid body as t=rFsinf.
REGAN PHY
Torque was defined previously for a rotating rigid body as t=rFsinf.
More generally, torque can be defined for a particle moving along
ANY PATH relative to a fixed point. i.e. the path need not be circular. z x y O f F
F redrawn
at origin
r x F
= t t

122. Angular Momentum z r x p = l l p redrawn at origin O x p y f
REGAN PHY
Angular Momentum z x y O f p
p redrawn
at origin
r x p
= l l

123. Newton’s 2nd Law in Angular Form.
REGAN PHY
Newton’s 2nd Law in Angular Form.

124. The net external torque, tnet acting on a system is equal to the rate
REGAN PHY
The net external torque, tnet acting on a system is equal to the rate
of change of the total angular momentum of the system ( L ) with time.

125. REGAN PHY z x y q w Dm

126. Conservation of Angular Momentum
REGAN PHY
Conservation of Angular Momentum

127. Example1: Pulsars (Rotating Neutron Stars)
REGAN PHY
Example1: Pulsars (Rotating Neutron Stars)
Crab nebula, SN remnant
observed by chinese in 11th century
before
after!
SN1987A
Pulsars have similar
periodicities ~0.1-1s.
Vela supernova
remnant, pulsar
period ~0.7 secs

128. Rotational period of crab nebula (supernova remnant) =1.337secs
REGAN PHY
Rotational period of crab nebula (supernova remnant) =1.337secs
Lighthouse
effect
Star
quakes
optical
PULSAR = PULSAting Radio Star (neutron-star)
x-ray

129. REGAN PHY
TRANSLATIONAL
ROTATIONAL

130. 13: Equilibrium and Elasticity
REGAN PHY
13: Equilibrium and Elasticity
An object is in ‘equilibrium’ if p=Mvcom and L about an any axis are
constants (i.e. no net forces or torques acts on the body).
If both equal to zero, the object is in STATIC EQUILIBRIUM.
If a body returns to static equilibrium after being moved (by a restoring
force, e.g., a marble in a bowl) it is in STABLE EQUILIBRIUM.
If by contrast a small external force causes a loss of equilibrium, it has
UNSTABLE EQUILIBRIUM (e.g., balancing pennies edge on).

131. The Centre of Gravity
REGAN PHY
The gravitational force acts on all the individual atoms
in an object. In principle these should all be added together
vectorially.
However, the situation is usually simplified by the concept
of the CENTRE OF GRAVITY (cog), which is the point
in the body which acts as though all of the gravitational force
acts through that point.
If the acceleration due to gravity, g, is equal at all points
of the body, the centre of gravity and the centre of mass
are at the same place.

132. Elasticity
REGAN PHY
A solid is formed when the atoms which make up the solid take up
regular spacings known as a LATTICE. In a lattice, the atoms take up
a repetitive arrangement whereby they are separated by a fixed, well
defined EQUILIBRIUM DISTANCE (of ~10-9->10-10m) from their
NEAREST NEIGHBOUR ATOMS.
The lattice is held together by INTERATOMIC FORCES which can
be modelled as ‘inter-atomic springs’. This lattice is usually extremely
rigid (i.e., the springs are stiff).
Note that all rigid bodies are however, to some extent ELASTIC.
This means that their dimensions can be changes by pulling, pushing,
twisting and/or compressing them. STRESS is defined as the
DEFORMING FORCE PER UNIT AREA= F/A,
which produced a STRAIN, which refers to a unit deformation.
The 3 STANDARD type of STRESS are (1) tensile stress ->DL/L
(stretching) ; (2) shearing stress -> Dx/L (shearing) ; and (3) hydraulic
stress -> DV/V (3-D compression).

133. STRESS and STRAIN are PROPORTIONAL TO EACH OTHER.
REGAN PHY
STRESS and STRAIN are PROPORTIONAL TO EACH OTHER.
The constant of proportionality which links these two quantities is
know as the MODULUS OF ELASTICITY, where
STRESS = MODULUS x STRAIN

134. F
REGAN PHY L L+ DL F Dx F L V DV
V-DV

135. If we plots stress as a function of strain,
REGAN PHY
If we plots stress as a function of strain,
for an object, over a wide range,
there is a linear relationship. This means
that the sample would regain its original
dimensions once the stress was removed
(i.e., it is ‘elastic’).
However, if the stress is increases BEYOND
THE YIELD STRENGTH, Sy,of the specimen,
it will become PERMANENTLY DEFORMED.
If the stress is increased further, it will ultimately
reach its ULTIMATE STRENGTH, Su, where the
specimen breaks/ruptures.
Strain (Dl/l)
Stress (F/A)
Sy (perm. deformed)
Su (rupture)

136. REGAN PHY
Example 1:
l =
81cm
F=62kN A
A cylindrical stainless steel rod has a radius r = 9.5mm and
length, L = 81cm. A force of 62 kN stretches along its length.
(a) what is the stress on the rod ?
(b) If the Young’s modulus for steel is 2.2 x 1011 Nm-2, what
are the elongation and strain on the cylinder ?

137. 14: Gravitation
REGAN PHY
Isaac Newton (1665) proposed a FORCE LAW which described the
mutual attraction of all bodies with mass to each other. He proposed
that each particle attracts any other particle via the
GRAVITATIONAL FORCE with magnitude given by
G=6.67x10-11N.m2/kg2=6.67x10-11m3kg-1s-2 is the gravitational constant
‘Big G’ (as opposed to ‘little g’ the acceleration due to gravity).
The two particles m1 and m2 mutually attract with a force
of magnitude, F. m1 attracts m2 with equal magnitude
but opposite sign to the attraction of m2 to m1. Thus,
F and -F form a third force pair, which only depends on
the separation of the particles, r, not their specific positions.
F is NOT AFFECTED by other bodies between m1 and m2. m1 m2 r
THE SHELL THEOREM:
While the law described PARTICLES, if the distances between the
masses are large, the objects can be estimated to be point particles.
Also, ‘a uniform, spherical shell of matter attracts a particle outside
the shell as if all the shell’s mass were concentrated at its centre’.

138. Gravitation Near the Earth’s Surface
REGAN PHY
Gravitation Near the Earth’s Surface
The earth can be thought of a nest of shells, and thus all its mass
can be thought of as being positioned at it centre as far as bodies
which lie outside the earth’s surface are concerned.
average ag at earth’s surface = 9.83 ms-2 altitude = 0 km
ag at top of Mt. Everest = 9.80ms-2 altitude = 8.8 km
ag for space shuttle orbit = 8.70 ms-2 altitude = 400km

139. We have assumed the free fall acceleration g equal the
REGAN PHY
We have assumed the free fall acceleration g equal the
gravitational acceleration, ag, and that g=9.8ms-2 at the earth’s surface,
In fact, the measured values for g differ. This is because
The earth is not uniform. The density of the earth’s crust varies. Thus
g varies with position at the earth’s surface.
The earth is not a sphere. The earth is an ellipsoid, flattened at the
poles and extended at the equator. (rpolar is ~21km smaller than requator).
Thus g is larger at poles since the distance to the core is less.
The earth is rotating. The rotation axis passes through a line joining
the north and south poles. Objects on the earth surface anywhere apart
these poles must therefore also rotate in a circle about this axis of
rotation (joining the poles), and thus have a centripetal acceleration
directed towards the centre of the circle mapped out by this rotation.

140. Centripetal Acceleration at Earth’s Surface
REGAN PHY
Centripetal Acceleration at Earth’s Surface N S w m R
mag N
‘above’ view,
looking from pole, R m w

141. Gravitation Inside the Earth
REGAN PHY
Gravitation Inside the Earth
‘A uniform shell of matter exerts no NET force on a particle located
inside it.’ R r m
Therefore, a particle inside a sphere only feels a
net gravitational attraction from the portion of
the sphere inside the radius at which it is at.
In the example on the left, for r = M/V = constant
a planet of radius, R and total mass M.
An object of mass m, which burrows downwards such that it is now at
a distance r from the centre of the planet (with r < R ).
The object will experience a gravitational attraction from the mass of
the planet inside the ‘shell’ of radius r and none from the portion of
the planet between radii r and the outer radius R. No
net
Force
Net
force

142. Gravitational Potential Energy
REGAN PHY
Gravitational Potential Energy
PROOF

143. Potential Energy and Force
REGAN PHY
Potential Energy and Force
Escape Speed (Velocity)

144. Johannes Kepler’s (1571-1630) Laws
REGAN PHY
THE LAW OF ORBITS: All planets move in elliptical orbits with the sun at one focus.
THE LAW OF AREAS: A line that connects a planet to the sun sweeps out equal areas in the plane of the planet’s orbits in equal times i.e., dA/dt=constant.
THE LAW OF PERIODS: The square of the period of any planet around the sun is proportional to the cube of the semi-major axis of the orbits. a b

145. The Law of Orbits If M >> m,the centre of mass of the planet-sun
REGAN PHY
If M >> m,the centre of
mass of the planet-sun
system is approximately
at the centre of the sun.
The orbit is described
by the length of the
semi-major axis, a
and the eccentricity
parameter, e.
The eccentricity is defined
by the fact that the each
focus f and f’ are distance
ea from the centre of the ellipse. A value of e=0 corresponds to a
perfectly circular orbit.
Note that in general, the eccentricities of the planetary orbits are small
(for the earth, e=0.0167). Rp is called the PERIHELION (closest
distance to the sun); Ra is the APHELION (further distance). f f’ r q Ra Rp a ea M m

146. REGAN PHY f f’ r q Ra Rp ea y r q f x f’ a

147. The Law of Areas
REGAN PHY Dq DA M m r

148. The Law of Periods
REGAN PHY r M m

149. Satellites, Orbits and Energies
REGAN PHY
Satellites, Orbits and Energies r
Energy
K(r)
Etot(r)
=-K(r)

150. REGAN PHY
Example 1:
A satellite in a circular orbit at an altitude of 230km above the earth’s
surface as a period of 89 minutes. From this information, calculate
the mass of the earth ?