1. Dynamics
Dynamics
Work/ Kinetic Energy Potential Energy
Conservative forces
Conservation laws
Momentum
Centre-of-mass
Impulse
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Part II – “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”
Chris Parkes
October

2. Work is the change in energy that results from applying a force
Work & Energy
Work is the change in energy that results from applying a force
Work = Force F times Distance s, units of Joules[J]
More Precisely, W=F.x
F,x Vectors so W=F x cos
Units (kg m s-2)m = Nm = J (units of energy)
Note 1: Work can be negative
e.g. Friction Force opposite direction to movement x
Note 2: Can be multiple forces, uses resultant force ΣF
Note 3: work is done on a specific body by a specific force (or forces)
The rate of doing work is the Power [Js-1Watts] F s F  x
So, for constant Force

3. in newtons acts on the particle.
Example
A particle is given a displacement
in metres along a straight line. During the displacement, a constant force
in newtons acts on the particle.
Find (a) the work done by the force and (b) the magnitude of the component of the force in the direction of the displacement.

4. 2 3 θ
F cos θ F
- 4 r
- 5

5. Work Done by Varying Force
Work-Energy Theorem
The work done by the resultant force (or the total work done) on a particle is equal to the change in the Kinetic Energy of the particle.
Meaning of K.E.
K.E. of particle is equal to the total work done to accelerate from rest to present speed
suggests
Work Done by Varying Force
W=F.x
becomes

6. Energy, Work Potential Energy, U Energy can be converted into work
Electrical, chemical, or letting a
weight fall (gravitational)
Hydro-electric power station
mgh of water
Potential Energy, U
In terms of the internal energy or potential energy
Potential Energy - energy associated with the position or configuration of objects within a system
Note: Negative sign

7. Gravitational Potential Energy Choice of zero level is arbitrary
Ug = mgh mg h
Reference plane
Ug = 0
- h mg
Ug = - mgh
No such thing as a definitive amount of PE mg
particle stays close to the Earth’s surface and so the gravitational force remains constant.

8. Stored energy in a Spring
This stored energy has the potential to do work Potential Energy
We are dealing with changes in energy h
choose an arbitrary 0, and look at  p.e.
This was gravitational p.e., another example :
Stored energy in a Spring
Do work on a spring to compress it or expand it
Hooke’s law
BUT, Force depends on extension x
Work done by a variable force

9. Work done by a variable force
Consider small distance dx over which force is constant
F(x)
Work W=Fx dx
So, total work is sum dx X F
Graph of F vs x,
integral is area under graph
work done = area dx X

10. Elastic Potential Energy
Unstretched position
For spring,F(x)=-kx: x F X X -X
Stretched spring stores P.E. ½kX2

11. Potential Energy Functionk
Reference plane x Fs mg

12. Conservation of Energy
K.E., P.E., Internal Energy
Conservative & Dissipative Forces
Conservative Forces
A system conserving K.E. + P.E. (“mechanical energy”)
But if a system changes energy in some other way (“dissipative forces”)
e.g. Friction changes energy to heat, reducing mechanical energy
the amount of work done will depend on the path taken against the frictional force
Or fluid resistance
Or chemical energy of an explosion, adding mechanical energy

13. Conservative forces
frictionless surface

14. Example
A 2kg collar slides without friction along a vertical rod as shown. If the spring is unstretched when the collar is in the dashed position A, determine the speed at which the collar is moving when y = 1m, if it is released from rest at A.

15. Properties of conservative forces
The work done by them is reversible
Work done on moving round a closed path is zero
The work done by a conservative force is independent of the path, and depends only on the starting and finishing points B
Work done by friction force is greater for this path A

16. Forces and Energy e.g. spring
Partial Derivative – derivative wrt one variable, others held constant
Gradient operator, said as grad(f)

17. Glider on a linear air track
Negligible friction
Minimum on a potential energy curve is a position of stable equilibrium
- no Force

18. Maximum on a potential energy curve is a position of unstable equilibrium

19. Linear Momentum Conservation
Define momentum p=mv
Newton’s 2nd law actually
So, with no external forces, momentum is conserved.
e.g. two body collision on frictionless surface in 1D
Also true for net forces
on groups of particles If
then
before m1 m2 v0
0 ms-1
Initial momentum: m1 v0 = m1v1+ m2v2 : final momentum
after m1 m2 v2 v1
For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components

20. Energy measured in Joules [J]
Energy Conservation
Energy can neither be created nor destroyed
Energy can be converted from one form to another
Need to consider all possible forms of energy in a system e.g:
Kinetic energy (1/2 mv2)
Potential energy (gravitational mgh, electrostatic)
Electromagnetic energy
Work done on the system
Heat (1st law of thermodynamics)
Friction  Heat
Energy measured in Joules [J]

21. Initial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E.
Collision revisited m1 m2 v2 v1
We identify two types of collisions
Elastic: momentum and kinetic energy conserved
Inelastic: momentum is conserved, kinetic energy is not
Kinetic energy is transformed into other forms of energy
Initial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E.
m1>m2
m1<m2
m1=m2
See lecture example for cases of elastic solution
Newton’s cradle

22. Impulse
Change in momentum from a force acting for a short amount of time (dt)
NB: Just Newton 2nd law rewritten
Where, p1 initial momentum
p2 final momentum
Q) Estimate the impulse
For Andy Murray’s serve
[135 mph]?
Approximating
derivative
Impulse is measured in Ns.
change in momentum is measured in kg m/s.
since a Newton is a kg m/s2 these are equivalent

23. Centre-of-mass Average location for the total mass
Mass weighted average position
Centre of gravity – see textbook
Position vector of centre-of-mass

24. dm is mass of small element of body
Rigid Bodies – Integral form y dm r x z
dm is mass of small element of body
r is position vector of each small element.

25. Momentum and centre-of-mass
Differentiating position to velocity:
Hence momentum equivalent to
total mass × centre-of-mass velocity
Forces and centre-of-mass
Differentiating velocity to acceleration:
Centre-of-mass moves as acted on by the sum of the Forces acting

26. Internal Forces Internal forces between elements of the body
and external forces
Internal forces are in action-reaction pairs and cancel in the sum
Hence only need to consider external forces on body
In terms of momentum of centre-of-mass

27. Example
A body moving to the right collides elastically with a 2kg body moving in the same direction at 3m/s . The collision is head-on. Determine the final velocities of each body, using the centre of mass frame.
4kg
6ms-1
3ms-1
2kg
C of M

28. Lab Frame before collision
4kg
6 ms-1
3 ms-1
5 ms-1
2kg
C of M
Centre of Mass Frame before collision
2 ms-1
1 ms-1
4kg
2kg
C of M
Centre of Mass Frame after collision
1 ms-1
2 ms-1
4kg
2kg
C of M