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

pdF

)(

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