MEI Conference 2007
Big ideas in Mechanics
Big ideas in mechanics
Journey into space
Big ideas in mechanics
Gravitation
Newton’s Laws
•
Newton’s Law of Universal Gravitation
and
•
Newton’s Laws of motion
work exactly
(almost)
Big ideas in mechanics
Universal gravitation
Newton’s Law of Universal Gravitation
Gm1m2
F=
2
d
•
m1
d
•
m2
Big ideas in mechanics
Dimensional analysis
2
Fd
G=
m1m2
−2
MLT × L
[G ] =
M×M
[G ] = M
−1 3
LT
2
−2
Big ideas in mechanics
Dimensionless constants
circumference
π=
diameter
L
[π ] =
L
π is dimensionless
Big ideas in mechanics
Fundamental dimensions
Some unanswered questions
• Can a “constant” with dimensions, like G and
the velocity of light, c, really be constant across
space and time ?
• What are the fundamental dimensions that
define the universe ?
• If we found such dimensions, would G and c
become dimensionless when referred to them ?
Big ideas in mechanics
Down to earth
Big ideas in mechanics
Life on earth
• Things don’t work exactly
• Even simple situations are complicated
• Nonetheless we make bridges that stay up and
aeroplanes that fly
• We use modelling to overcome the difficulties
Big ideas in mechanics
Modelling
A problem
Simplify it to allow
work to begin
Use mathematics
to solve it
Is
the answer
accurate enough
?
Yes
Problem solved
Review
simplification
No
Big ideas in mechanics
Simplifications
•
•
•
•
•
•
•
Particle
Light
Smooth
Inextensible
Perfectly elastic
Linear
Rigid
Big ideas in mechanics
Friction
What happens if a surface is not smooth ?
R
μ
≤
F
Refined model
Is this model accurate ?
Big ideas in mechanics
Gravitation
Is Newton’s Law of Universal Gravitation
absolutely accurate ?
• Counter-example
The orbit of Mercury
• Refined model
General relativity
Big ideas in mechanics
Reducing simplification (1)
What are the implications of reducing the
simplification in the cases listed before ?
• Particle model → Rigid body model (Moments)
• Light → Heavy (Centre of mass)
• Smooth → Rough (Friction)
• Inextensible → Elastic (Hooke’s Law)
Big ideas in mechanics
Reducing simplification (2)
What are the implications of reducing the
simplification in the cases listed before ?
• Perfectly elastic → Not perfectly elastic
(Restitution)
• Linear motion → General motion (Rotation,
moments of inertia)
• Rigid → Flexible (Bending moments)
Big ideas in mechanics
Newton’s 1st Law of Motion
Every particle remains in a state of rest or
uniform motion in a straight line unless acted
on by a resultant external force
• This is the source of common misconceptions
about Force; for many students it is counterintuitive and does not match their experience
- to hold a brick up in the air you need a force
- to push a box along a level floor at steady
speed you need a force
Big ideas in mechanics
Newton’s 2nd Law of Motion
The rate of change of linear momentum of a
particle is proportional to the resultant force
and is in the direction of the force. … When the
mass is constant, the law becomes F= ma
• This law states the relationship between force
and acceleration; it links dynamics (forces) and
kinematics (motion)
Big ideas in mechanics
Newton’s 3rd Law of Motion
When one object exerts a force on another there
is always a reaction which is equal, and
opposite in direction, to the acting force
• Misconceptions about reaction forces are
common.
- Students should be strongly encouraged to
draw appropriate force diagrams.
Big ideas in mechanics
Kinematics
Powerful ideas for students are carried by
simple kinematics
• The quantities v, s, a and t are linked without
the need to know what causes the motion
• They are linked with dynamics through
Newton’s Laws
• Calculus is put to good use
Big ideas in mechanics
Vectors
For those who understand them vectors
are liberating as they allow us to
manipulate size and direction together
Newton’s laws are about vector
quantities and so may be applied in any
direction
Big ideas in mechanics
Systems
Big ideas in mechanics
Connected systems
• A system can be analysed as separate parts
• Each part is separately subject to the forces
acting upon it so that it is separately in
equilibrium or has acceleration related to that
of the whole system
• Newton’s 3rd Law applies at the interfaces of
the parts
Big ideas in mechanics
Conservation
Laws of conservation of:
•
•
•
•
Matter
Energy
Linear momentum
Angular momentum
Big ideas in mechanics
Energy and momentum
Many students do not separate the
concepts of energy and momentum in
their minds
• How can you help them ?
• What examples or images can you give
them ?
Big ideas in mechanics
Momentum and energy
Asteroid Mass 1000 kg, speed 1.5×104 ms-1
Supertanker: Mass 5×108 kg, speed 10 ms-1
Big ideas in mechanics
Angular momentum
Big ideas in mechanics
Fundamental laws
•
•
•
•
Which of the conservation laws are really
fundamental ?
Matter
Energy
Linear momentum
Angular momentum
Are these the fundamental dimensions of
the universe ?
Big ideas in mechanics
Why teach mechanics ?
Give me a place to stand and I will move the
earth
Archimedes