Post on 01-Jan-2016
description
Dynamics
Chris ParkesOctober 2013
Dynamics
Work/ Kinetic Energy Potential Energy
Conservative forces
Conservation laws
Momentum
Centre-of-mass
Impulse
http://www.hep.manchester.ac.uk/u/parkes/Chris_Parkes/Teaching.html
Part II – “We are to admit no more causes of natural
things than such as are both true and
sufficient to explain their appearances.”
READ the Textbook!
Work & Energy
• 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-1Watts]
Work is the change in energy that results from applying a force
Fs
x
F
dt
dWP So, for constant Force
Example
A particle is given a displacement jir ˆ5ˆ2
in metres along a straight line. During the displacement, a constant force
jiF ˆ4ˆ3
Find (a) the work done by the force and (b) the magnitude of the component of the force in the
direction of the displacement.
in newtons acts on the particle.
r
F
θ
F cos θ
32
- 4
- 5
Work-Energy TheoremThe 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 ForceW=F.x becomes
Energy, Work• Energy can be converted into work
– Electrical, chemical, or letting a
weight fall (gravitational)• Hydro-electric power station
mgh of water
FdxUUW
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
Potential Energy, U
Reference plane
mg
mg
mg
Ug = 0
h
- h
Ug = mgh
Ug = - mgh
Gravitational Potential Energy
particle stays close to the Earth’s surface and so the gravitational force remains constant.
No such thing as a definitive amount of PE
Choice of zero level is arbitrary
This stored energy has the potential to do work Potential EnergyWe are dealing with changes in energy
0h
• choose an arbitrary 0, and look at p.e.
This was gravitational p.e., another example :
Stored energy in a SpringDo 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
hmgxFW )(
Work done by a variable forceConsider small distance dx over which force is constant
F(x)
dx
Work W=Fx dx
So, total work is sum
0 X
X
dxxFdxFW0
)(
Graph of F vs x,
integral is area under graph
work done = area
F
Xdx
Elastic Potential EnergyUnstretched position
X
-X
221
02
21
00
][)( kXkxkxdxdxxFW XXX
For spring,F(x)=-kx:
Fx
X
Stretched spring stores P.E. ½kX2
Potential Energy Function
Reference plane
k
x
mg
Fs
• 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
Conservative & Dissipative Forces
Conservation of Energy
K.E., P.E., Internal Energy
Conservative forces
frictionless surface
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.
Properties of conservative forces
• Work done on moving round a closed path is zero
Work done by friction force is greater for this path
• The work done by a conservative force is independent of the path, and depends only on the starting and finishing points
• The work done by them is reversible
A
B
Forces and Energy FdxUUW
e.g. spring
• Partial Derivative – derivative wrt one variable, others held constant• Gradient operator, said as grad(f)
Minimum on a potential energy
curve is a position of stable
equilibrium
- no Force
Glider on a linear air track
Negligible friction
Maximum on a potential energy curve is a position of unstable equilibrium
U
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
amdt
vdm
dt
vmd
dt
)(
before
after
m1 m2
m1 m2
v0 0 ms-1
v1v2
For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components
Initial momentum: m1 v0 = m1v1+ m2v2 : final momentum
constpdt
pdF ,0,0 Also true for net forces
on groups of particlesIf
then constpp
FF
ii
ii
,0
Energy Conservation
• 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 can neither be created nor destroyed
•Energy can be converted from one form to another
Energy measured in Joules [J]
Collision revisited• 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 m2v2
2 : final K.E.
m1 v1
m2 v2
See lecture example for cases of elastic solution 1. m1>m2
2. m1<m2
3. m1=m2
Newton’s cradle
Impulse• Change in momentum from a force acting
for a short amount of time (dt)
• NB: Just Newton 2nd law rewritten
dtFppJ 12
Impulse Where, p1 initial momentum p2 final momentum
amdt
vdm
dt
pd
dt
ppF
12
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
Q) Estimate the impulseFor Andy Murray’s
serve [135 mph]?
Centre-of-mass
• Average location for the total mass
• Position vector of centre-of-mass
Mass weighted average positionCentre of gravity – see textbook
dmr
x
y
z
Rigid Bodies – Integral form
dmrM
r cm
1
dm is mass of small element of bodyr is position vector of each small element.
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
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
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.
4kg6ms-1
2kg3ms-1
C of M
4kg6 ms-1
2kg3 ms-1
C of M
Lab Frame before collision
Centre of Mass Frame before collision
5 ms-1
4kg1 ms-1
2kg2 ms-1
C of M
Centre of Mass Frame after collision
4kg1 ms-1
2kg2 ms-1
C of M