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Introduction to Mechanics

01 Modelling

Introduction to Mechanics
01 Modelling
In mechanics it is necessary to take a real life problem and put it in mathematical language
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Terminology
When modelling, we often make assumptions to make mathematics simpler:
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A particle is a body whose entire weight acts through a single point
A particle doesn’t suffer from resistance
A lamina is a two dimensional body, it is the 2-D equivalent of a particle
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It is impossible to stretch
A rigid body is a body which is not a point and whose shape is fixed

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Introduction to Mechanics

02 Acceleration

02 Acceleration
The “SUVAT” equations
Acceleration is the rate of change of velocity of an object
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This equation can be rearranged to give:

𝒗 = 𝒖 + 𝒂𝒕

If

𝒔 represents displacement of an object then:

𝒔=

𝟏
(𝒖 + 𝒗)𝒕
𝟐

𝒔 = 𝒖𝒕 +

𝟏 𝟐
𝒂𝒕
𝟐

𝒗 𝟐 = 𝒖 𝟐 + 𝟐𝒂𝒔
These equations are true if the acceleration of the body in question is constant (i
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it doesn’t change
over the time interval)
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The acceleration caused by gravity
is written as

𝒈 and is usually taken to be 9
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The ball is dropped from rest
and falls freely under gravity
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𝟖, 𝒖 = 𝟎

Therefore the correct equation is:

𝒔 = 𝒖𝒕 +

𝟏
𝟐

𝒂𝒕 𝟐

𝟓𝟎 = 𝟎 × 𝒕 +

𝒕𝟐 =

𝟏
𝟗
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𝟖
𝟐

𝒕 𝟐 = 𝟏𝟎
...
𝟏𝟗 𝒔

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Introduction to Mechanics

03 Newton's Laws of Motion

03 Newton's Laws of Motion
Forces
A force is an influence tending to cause the motion of a body
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This means that in order for the acceleration of a body to change there must be a net force applied
to the body
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So if we are told that the body is not accelerating (i
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if it is
moving at a constant velocity), we know that the overall (resultant) force in any one direction will be
zero
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The body moves at a constant speed of 5 ms-1
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It should be clear that

𝒙 = 𝟓
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Introduction to Mechanics

03 Newton's Laws of Motion

Newton’s Second Law of Motion
This states that the rate of change in momentum of the body is directly proportional to the net force
applied
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By how
much the acceleration changes depends upon the magnitude of the force applied
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Mass is the amount of matter a body
contains and is measured in kilograms (kg)
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Newton’s Third Law of Motion
This states that every action has an equal and opposite reaction
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At the same time, however, the table exerts a force on
the ball
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Introduction to Mechanics

04 Vectors

04 Vectors
A Vector quantity has both magnitude and direction
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A scalar quantity has only magnitude, so the direction is not important
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Vectors are usually represented as follows:

The arrow shows the direction and the number ( 𝒗 in this case) represents the magnitude
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Unit Vectors
A unit vector is a vector which has a magnitude of 1
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The unit
𝒋 and for the 𝒛 axis is 𝒌
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If the vectors are given unit form simply add together the 𝒊, 𝒋,

𝒑 = 𝟑𝒊 + 𝒋,

𝒒 = −𝟓𝒊 + 𝒋,

𝒌 values, for example:

𝒇𝒊𝒏𝒅 𝒑 + 𝒒

∴ 𝟑𝒊 + 𝒋 + −𝟓𝒊 + 𝒋 = −𝟐𝒊 + 𝟐𝒋

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Introduction to Mechanics

04 Vectors

This could also be worked out by diagram:

The Magnitude of a Vector
A vectors’ magnitude can be found using Pythagoras’s Theorem:

𝑴𝒂𝒈𝒏𝒊𝒕𝒖𝒅𝒆 𝒐𝒇 𝒂𝒊 + 𝒃𝒋 = √( 𝒂 𝟐 + 𝒃 𝟐 )

Resolving a Vector
This means finding its magnitude in a particular direction
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Using basic trigonometry, we can calculate
that the component of 𝒓 in the direction of the 𝒙 axis is 𝒓 𝒄𝒐𝒔 𝒋
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Therefore:
In the diagram the vector

𝒓 = 𝒓 𝒄𝒐𝒔 𝒋𝒊 + 𝒓 𝒔𝒊𝒏 𝒋𝒋

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Introduction to Mechanics

05 Using Newton's Laws

05 Using Newton's Laws
Newton’s laws of motion can be used to work out forces and acceleration of bodies
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Mark on all of the forces and
acceleration
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It is connected by a light inextensible string,
which passed over a smooth pulley at the edge of the table, to another body of mass 3 kg which is
hanging freely
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Find the tension in the string and the acceleration
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If the body hanging freely accelerates at a certain rate, the string will pull the
other body so that it accelerates at the same rate
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This is because the
pulley is smooth
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The mass of the first body is 5 kg and so its weight is 5g N
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Remember to mark in the normal reaction force
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𝑻 = 𝟓𝒂
𝑻=
𝑻=

Therefor the acceleration:

𝟓 (𝟑𝒈)
𝟖
𝟏𝟓𝒈
𝟖

=

𝟑(𝟗
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𝟖𝟏)
𝟖

And the tension:

= 𝟏𝟖
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The moment of the force

𝑭 about point 𝑨 is fixed
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In the above example the moment is clockwise
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Couples
A couple is a system of forces which has a net moment but does not have a resultant force in any
one direction
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If a system is in equilibrium, the system will have zero resultant force and the sum of the moments
about any point will be zero
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Momentum is measured in
Ns (newton second)
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This is also equal to the magnitude of the force multiplied by the length of time the
force is applied

𝑰𝒎𝒑𝒖𝒍𝒔𝒆 = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎 = 𝑭𝒐𝒓𝒄𝒆 × 𝒕𝒊𝒎𝒆

Conservation of Momentum
When there is a collision between two objects, Newton’s Third Law states that the force on one of
the bodies is equal and opposite to the force on the other body
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Hence the total momentum before the collision in a particular
direction must equal the total momentum in a particular direction after collision
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Introduction to Mechanics

07 Impulse and Momentum

𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎 = (𝟑 × 𝟑) − (𝟐 × 𝟏) = 𝟕
𝑭𝒊𝒏𝒂𝒍 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎 = (𝟐 × 𝟏) − (𝟑𝒙)
𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎 = 𝑭𝒊𝒏𝒂𝒍 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎
∴ 𝟕 = 𝟐 − 𝟑𝒙
𝒙=

−𝟓
𝟑

N
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Velocity of the balls was relative to the 3 kg ball in the first diagram i
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left to right is a positive
velocity, right to left is a negative velocity
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If it were travelling in the opposite

𝟑𝒙
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Introduction to Mechanics

08 The Coefficient of Friction

08 The Coefficient of Friction
Friction is the force or resistance an object encounters in moving over another
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Applying a small force will not cause the book to move
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If the frictional force
were less that the applied force, the book would move
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If pushing the book harder, it remains stationary, the frictional force must have increased, or the
book would have moved
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When the frictional force is at its maximum possible value, friction is said to be
limiting
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Pushing slightly harder, the book will move
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The frictional force between two objects is not constant but increases until it reached its maximum
value
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The Coefficient of Friction
This is a number which represents the friction between two surfaces
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The symbol used for the coefficient of friction is

𝒎
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The frictional force, 𝑭, will act parallel to the surfaces in contact and in a direction opposite the
motion that is taking or trying to take place
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Find the coefficient of friction between the particle and the plane
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𝟓𝟕𝟕
𝒄𝒐𝒔 𝟑𝟎

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