Vector and scalar quantities
• A scalar quantity is defined only by its magnitude (or size)
for example: distance, speed, time ….
• A vector quantity is defined by stating both its magnitude
(or size) and direction for example: displacement, velocity,
force……
It is easy to combine two or more
scalar quantities e.g.
2 metres + 3 metres = 5 metres!
The direction must be taken into
account when combining two or
more vector quantities
e. 2 metres + (-3 metres) = -1 metre
Introduction to
Forces II
Sharon Tripconey
Some Basic PrinciplesNewton’s first law (N1L)
• Every particle continues in a state of rest or uniform motion in a straight line unless acted on by a resultant external force.
This means that for a particle to be in equilibrium* it must be the case that there is no resultant force acting on it (*dynamic or static equilibrium)
Newton’s second law (N2L)
• When a force acts on a particle, the change in momentum is proportional to the force. For constant mass, F = ma
Newton’s third law (N3L)
• When one object exerts a force on another there is always a reaction that is equal in magnitude and opposite in direction to the applied force.
This means that we expect forces to be found in ‘pairs’
Draw a force diagram
Assumptions:
•Each object is a particle in equilibrium
•The scales and the brush have zero mass
W
W = Weight of the person
Hint: Think about pairs of
forces (N3L)
W
F
F
S
SR
R
N
N
W Weight of the person
F Force exerted on the brush
S Contact force (brush and floor)
N Contact force (person and scales)
R Contact force (scales and floor)
W
Problem:
Explain why the reading goes down when I press on the floor
with a brush
F has the same magnitude as
the force exerted on the brush
N is the contact force (the reading on
the bathroom scales)
Consider the forces acting
on the person
F + N – W = 0+
F + N = W
W is the weight of the person (which
is constant)
Newton’s Second LawIn a given direction F = ma
Example
A toy train of mass 0.5 kg moves on a horizontal
straight track. There is a driving force of 0.4 N and
a resistance to motion of 0.35 N. Find the
acceleration.
0.4 N0.35 N
a m s -2
0.5 kg
Newton’s Second Law
The acceleration is 0.1 ms -2 in the direction of the 0.4 N force.
2 Resultant force = ma
0.4 0.35 0.5
0.05 = 0.5
0.1
N L
a
a
a
0.4 N0.35 N
a m s -2
0.5 kg
Two identical objects are connected to the ends of a light
inelastic string which passes over a fixed pulley as shown
What happens if the system is released from rest?
What happens if….
Motion in a lift
A boy of mass 50 kg is standing in a lift of mass 200 kg. The lift is raised by a vertical cable. Find the reaction of the floor on the boy and the
tension in the cable when the acceleration is 1
4𝑔 upwards.
T
R 200g
¼ g
lift and boy (250kg)boy (50kg) lift (200kg)
¼ g ¼ g
50g
R T
250g
Motion in a lift
14
50 50 so 62.5R g g R g
14
200 200 so 312.5T R g g T g
For the lift and boy, N2L ↑,
𝑇 − 250𝑔 = 250 ×1
4𝑔 so 𝑇 = 312.5𝑔
For the boy, N2L ↑
For the lift, N2L ↑,
• A model is a representation of a real situation.
A real situation will invariably contain a rich
variety of detail and any model of it will simplify
reality by extracting those features which are
considered to be most important.
• Modelling is at the heart of the subject of
Mechanics
Mathematical modelling
Modelling assumptions• Common assumptions are:
• the body is a particle
• value of g is constant
• the string is light and inextensible
• the pulley is light and smooth
• air resistance is negligible
• friction is negligible
• friction obeys the law F µ R
Modelling Cycle
Real-world
problem
Simplifying
assumptions
Mathematical
model
(equations etc)
Analysis
and solutionPredictionExperiment
How can we support students with their
understanding of mathematical modelling in
Mechanics?
• Incorporate simple practical demonstrations into our teaching;
many students quickly see something of the principle even if
their depth of understanding is not great
• As teachers, knowledge of common student misconceptions
helps us to plan lessons that will reduce the likelihood of their
development and increase the likelihood that misconceptions
already formed will be corrected.
• An approach of confronting hard things is usually more
successful than that of avoiding them for as long as possible!
About MEI• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students and
work with employers to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources
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