Chapter 2 - Mechanics

&
Chapter 6: Circular Motion and Gravitation
broly#6936
January 25, 2021

Contents
1 Motion 2
1.1 Distance and Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Speed and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Graphs Describing Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Equations of Uniformly Accelerated Motion . . . . . . . . . . . . . . . . . . . . 3
1.6 Fluid Resistance and Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . 3
1.7 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7.1 Projectile Motion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7.2 Projectile Motion Summary . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.8 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8.1 Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8.2 Centripetal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Forces 8
2.1 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Linear Momentum and Impulse . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Law of Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Elastic Collisions, Inelastic Collisions and Explosions . . . . . . . . . . . 9
2.3 Work, Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.2 Principle of Conservation of Energy . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Gravitational Field Strength . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Gravitational Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.3 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.1 Source of Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.2 Kepler’s Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.3 Weightlessness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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1 Motion
The average velocity during an interval of time ∆t is the ratio of the change in position ∆s
during that time interval to ∆t.

In uniform motion, there is no acceleration, therefore; velocity is constant so the term ‘average’
is unnecessary. The velocity is the same at all times.
s − s0
v= −→ s = s0 + vt
t − t0

1.1 Distance and Displacement

ˆ Distance: Length of path followed; scalar quantity → s.

ˆ Displacement: Change in position; vector quantity → ~s = s − s0 .

1.2 Speed and Velocity

ˆ Speed: Rate of change of distance with respect to time → v = ds
dt

ˆ Velocity: Rate of change of displacement with respect to time → ~v = d~s

dt
.

1.3 Acceleration
ˆ Acceleration: Rate of change of velocity with respect to time → ~a = dv
dt
= ~v −~
t
u

1.4 Graphs Describing Motion

ˆ d
dt
~s(t) = ~v (t) → d
dt
~v (t) = ~a(t)

ˆ
R R
~a(t)dt = ~v (t) → ~v (t) = ~s(t)

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1.5 Equations of Uniformly Accelerated Motion

(v + u)t
v = u + at s=
2
1
v 2 = u2 + 2as s = ut + at2
2

1.6 Fluid Resistance and Terminal Velocity

ˆ The motion of objects through the air is opposed by the force of air resistance. Similar
forces arise when any object moves in any direction through any fluid and generally such
forces are described as fluid resistance or drag.

ˆ An object which is able to move through a fluid with low resistance may be described as
streamlined.

ˆ The amount of fluid resistance acting on any moving object depends on its speed, its
cross-sectional area and shape.

ˆ When fluid resistance becomes equal and opposite to the weight, a falling object will
reach a constant velocity, known as terminal velocity.

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1.7 Projectile Motion
Any unpowered object moving through the air will follow a trajectory, affected by the strength
of the gravitational field (and if significant, air resistance). Such objects are often called pro-
jectiles.

1.7.1 Projectile Motion Analysis

1. Calculate the Components of Velocity

ˆ Horizontal Velocity: vx = v · cos(θ)

ˆ Vertical Velocity: vy = v · sin(θ)

If the vertical velocity component is equal to 0, then it’s the case of horizontal projectile
motion. If, additionally, θ = 90° then it’s the case of free fall.

2. Write down the equations of motion

ˆ Distance
– Horizontal distance travelled can be expressed as:

x = vx t

– Vertical distance from the ground can be expressed by the equation:

1
y = y0 + vy t − gt2
2
ˆ Velocity
– Horizontal velocity: vx
– Vertical velocity: vy · gt
ˆ Acceleration
– Horizontal acceleration: ax = 0
– Vertical acceleration: ay = −g

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3. Calculate the time of flight

ˆ Flight ends when the projectile hits the ground. We can say that it happens when
the vertical distance from the ground is equal to 0. In the case where the initial
height is 0, the equation can be written as:
1
vy − gt2 = 0
2
Then, from this equation, we find that the time of flight is:

2vy 2v sin(θ)
t= =
g g

ˆ However, if we’re throwing the object from some elevation, then the formula is not
so nicely reduced as before, and we obtain a quadratic equation to solve:
1
y0 + vy t − gt2 = 0
2
After solving this:
p 2 p
vy + vy + 2gy0 v sin(θ) + (v sin(θ))2 + 2gy0
t= =
g g

4. Calculate the range of the projectile

ˆ The range of the projectile is the total horizontal distance traveled during the flight
time. Again, if we’re launching the object from the ground (initial height = 0), then
we can write the equation as:
2vy
x = vx t = vx ∗
g
This transforms to:
v 2 · sin(2θ)
x=
g
ˆ Things are getting more complicated for initial elevation differing from 0. Then, we
need to substitute the long equation from the previous step as t:
p
vy + vy2 + 2gy0
x = vx t = vx ·
g

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 5. Calculate maximum height

ˆ When the projectile reaches the maximum height, it stops moving up and starts
falling. It means that its vertical velocity component changes from positive to neg-
ative – in other words, it is equal to 0 for a brief moment at time: vy (t) = 0

If:
vy − gt(vy = 0) = 0
This becomes:
vy
thmax =
g
ˆ Now, we simply find the vertical distance from the ground at that time:

1 vy2 v 2 · sin2 (θ)

ymax = vy tmax − gt2max = =
2 2g 2g

ˆ Fortunately in the case of launching a projectile from some initial height y0 , we need
to simply add that value into the final equation:

v 2 sin2 (θ)
ymax = y0 +
2g

1.7.2 Projectile Motion Summary

ˆ Horizontal Velocity Component: vx = v · cos(θ)

ˆ Vertical Velocity Component: vy = v · sin(θ)

ˆ Launching the object from the ground (initial height y0 = 0)

2vy
– Time of Flight: t = g
2vx ·vy
– Range: x = g
vy2
– Maximum Height: hmax = 2g

ˆ Launching the object from some elevation (initial height y0 > 0)

√
vy + vy2 +2gy0
– Time of Flight: t = g
√2
vy + vy +2gy0
– Range: x = vx · g
vy2
– Maximum Height: ymax = y0 + 2g

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1.8 Uniform Circular Motion
Uniform circular motion refers to circular motion at constant speed.
In a uniform circular motion, speed is constant while (angular) velocity and (angular) acceler-
ation are constantly changing.

ˆ While the magnitude of its velocity remains constant, the direction of its velocity is
constantly changing.

ˆ The acceleration causing this change in velocity is always directed towards the center of
the circular path (radial acceleration).

1.8.1 Centripetal Force

Centripetal force is the corresponding force (resultant force) which causes the centripetal ac-
celeration.

ˆ Direction: Pointing towards the center of the circle / perpendicular to the instantaneous
velocity

ˆ Magnitude:
mv 2
F~c =
r
ˆ Work done by centripetal force = 0

1.8.2 Centripetal Acceleration

The acceleration which gives rise to a circular motion is called the centripetal acceleration. Its
magnitude is given by:
v2
~ac = ω 2 r =
r
Where ω is angular velocity, v is linear velocity (v = ωr = 2πr
T
) and r is the radius.
2π 4π 2 r
Since v = T , ac can also be written as: ~ac = T 2

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2 Forces
2.1 Newton’s Laws of Motion
ˆ Newton’s First Law - Law of Inertia:

When no external forces act on a body ( F~net = 0) the body will either remain at rest
P
(v = 0ms−1 ) or continue to move with constant speed a = 0ms−2 .

ˆ Newton’s Second Law:

Force is equal to the product of mass and acceleration:

F~net = m~a

Hence when the change in velocity is equal to zero, the object will either remain at a
constant speed or remain at rest.
d~p d~v
F~ = =m
dt dt

ˆ Newton’s Third Law:

For every action, there is an equal and opposite reaction;

F~AB = −F~BA

If body A, exerts a force on body B, then body B will exert and equal but opposite force
on body A. These two resultant forces will have an equal magnitude but and opposite
direction and the force will be exerted on different body’s.

2.2 Momentum
2.2.1 Linear Momentum and Impulse
Newtons original second law:
d~p
F~ =
dt
Linear Momentum: Mass times linear velocity

p~ = m · ~v

Impulse: Change in momentum

d~p
I~ = = ∆~p = F~ ∆t
dt

2.2.2 Law of Conservation of Momentum

For a collision occurring between object 1 and object 2 in an isolated system, the total momen-
tum of the two objects before the collision is equal to the total momentum of the two objects
after the collision.

p~i = p~f

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 mi · ~v0 = mf · ~vf
For 2 objects, 1 and 2, colliding:

p~1,i + p~2,i = p~1,f + p~2,f

Momentum is ALWAYS conserved.

2.2.3 Elastic Collisions, Inelastic Collisions and Explosions

ˆ A collision in which the sum of the kinetic energies of all masses is the same after the
collision as it was before is known as an elastic collision.

ˆ An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic

energy is not conserved due to the action of internal friction.

ˆ When colliding objects stick together, it is described as a totally inelastic collision.

2.3 Work, Energy and Power

2.3.1 Work
Work is not energy, it is a means of transferring energy by a force applied to a moving object.
If the object does not move or the force is not in the direction of the motion then the force
is not transferring energy to the object or we say “the force is not doing work on the object.”
Mathematically we define work as:

W = F~ • s = F · s · cos(θ)

W = ∆EK = −∆EP

Unit: Joules (J = kg · m2 s−2 )

An object can also have elastic potential energy, if a spring is stretched or a rubber ball is
deformed. . . In the case of the a spring the force of the spring is given by Hooke’s Law:

F = kx
So the work done is by displacing the spring a distance x is:

W = kx2

2.3.2 Principle of Conservation of Energy

Energy conservation is the principle that in a closed system energy is neither gained or lost. In
an open system it is possible for energy to be added or lost. This definition can also be turned
around. If a system is gaining or losing energy then it is an open system. The only truly open
system in the universe, the earth is constantly gaining energy from the sun during the day and
radiating the energy back out at night.

E0 = Ef
In terms of potential and kinetic energy:

EK0 + EP 0 = EKf + EP f

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2.3.3 Power
Power is the rate that work is done or the rate that energy is transferred.
W Fs
P = = = Fv
t t
Unit: Watts (W = Nms−1 )

2.3.4 Efficiency
When work is done on an object, sometimes the energy is converted into unwanted or non-useful
forms (often heat). The ratio of useful energy to the amount of energy applied is the efficiency,
it can also be defined in terms of power:

Useful Power / Energy

Efficiency(η) =
Total Power / Energy

2.4 Gravitation
The force of attraction is gravity, the same force that holds us on the ground and makes things
falls. Newton supposed that gravity acts everywhere and can be described by the following
equation:
Mm
F~g = G 2
r
Where M and m are the masses of the two objects, r is the distance between them, and G is
the universal gravitation constant: G = 6.67 × 10−11 Nm2 kg−2

2.4.1 Gravitational Field Strength

The gravitational field strength at a point is the force per unit mass experienced by a test mass
at that point.
The gravitational field strength (g) due to an object is given by:

GM FG
g=− 2
=
r m
If the mass is moved the work done on the mass is equal to the change in potential energy
(assuming the movement is slow so as not to add kinetic energy). Work is defined as , however
the force is not constant and varies with distance.
Mm
−W = G = −∆EP = ∆EK
r
Note: That the work done is negative. This is because the force and displacement are in
opposite directions. This also makes gravitational potential energy negative. From this, we can
say that two objects with mass separated by a distance r have potential energy:
Mm
EP = −G
r

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2.4.2 Gravitational Potential
The gravitational potential is a field but it is a scalar not a vector. One way of thinking of
the gravitational potential is that it is the gravitational potential energy per unit mass. The
gravitational potential due to a point mass (or uniform sphere) can be described mathematically
as:
EP GM
V = =−
m r
Unit: Volts (V = Jkg−1 )

2.4.3 Escape Velocity

When a mass is thrown up, with a low initial velocity, it falls back down. From an energy
argument, this could be said to be a result of the object having a non-zero (or negative)
potential energy, there is potential energy that tries to make the mass fall back down. But if
an object could have zero gravitational potential energy it would never fall back down. The
total energy of a mass m, where the only force acting is gravity due to a second mass M, can
be described as:
1 Mm
ET = EK + EP = mv 2 − G
2 r
If we want to send a satellite out beyond the earth or beyond the solar system, for exploration
or research, we basically need to be able to throw it up with out it coming back down. In order
for something to not come back down it needs to be “free of gravity.” As mentioned earlier if
an object is moved to ”infinity” it will have no gravitational potential energy.

Since gravitational potential energy is always non-positive (zero or negative), this means that
for an object to be free of gravity it must have either zero or positive total energy. In an
equation this can be written as:
1 Mm 1 2
E = mv02 − G = mv∞ ≥0
2 r 2
Where v0 is the objects initial velocity and v∞ is the objects velocity once it reaches “infinity”

If an object is to just barely escape gravity, that is to get to ”infinity” and have no velocity,
the total energy must be equal to zero. If we then solve for the initial velocity:
1 2 Mm
mve − G =0
2 r
r
2M
ve = G
r
This velocity is known as the escape velocity, it is the minimum velocity for an object to escape
the gravitational pull of another mass M. Note that the escape velocity is independent of the
objects mass. The escape velocity for the earth is approximately 11kms−1

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2.5 Orbital Motion
2.5.1 Source of Centripetal Force
If you attach a ball to the end of a string and swing the ball around the ball moves in a circle, no
surprise. The string provides a centripetal force to keep the ball moving in a circle as opposed
to moving in a straight line (Newton’s 1st Law). For a planet orbiting a star or a satellite
orbiting the earth, gravity provides this centripetal force.

Assuming that the path of the satellite is circular we can use the Newton’s law of Universal
Gravitation:
Mm
F =G 2
r
and Newton’s second law:
v2
F = ma = m
r
To calculate the speed of an orbiting satellite:
r
Mm v2 M
G 2 = m −→ v = g
r r r
Note that the velocity is independent of the mass of the satellite. The above equation will work
for any planet or star (or any an object that has something in orbit) with mass M and radius r.
If we want to calculate the velocity for a object orbiting the earth we can use the knowledge:
Me p
g=G ∴ v= gRe
Re2

2.5.2 Kepler’s Third Law

T 2 ∝ R3
Kepler made this assumption with no knowledge of the Law of Universal Gravitation. Kepler’s
3rd Law can be easily derived, if T is the period (time for one revolution) then we can say:
2πR
v=
T
Where R is the radius of the orbit. Using the result we found for orbital velocity above:
 2πR 2 GM
=
T R
4π 2 R3 = GM T 2
4π 2 3
T2 = R
GM

2.5.3 Weightlessness
What does it mean to be weightless? Maybe a better first question is what does it mean to be
weightful?

If you stand on a scale why does the scale register a reading? There are two sets of action and
reaction force, one between you and the scale and another between the scale and the ground.
When you stand on the scale the ground pushes up on the scale that allows the scale to push
up on you and complete the action reaction pair. If this action reaction pair doesn’t exist the

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scale can’t give a reading.

During free fall the “ground” or surface that a scale would rest on is accelerating at the same
rate as the scale, therefore the surface can not exert a force on the scale and therefore the scale
would read zero. This is what is meant by weightlessness. You are not free of gravity, you
simply can not measure your weight.

During orbital motion an object is continuously in free fall, it’s falling around the planet or
whatever object it is orbiting. So when orbiting an object is weightless.

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