Dynamics

Chris ParkesOctober 2013

Dynamics

Velocity & Acceleration

Inertial Frames

Forces – Newton’s Laws

http://www.hep.manchester.ac.uk/u/parkes/Chris_Parkes/Teaching.html

Part I - “I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a

hypothesis; and hypotheses, whether metaphysical or physical,

whether of occult qualities or mechanical, have no place in experimental philosophy.”

READ the Textbook!

vector addition• c=a+b

cx= ax +bx

cy= ay +by

scalar product

x

y

a

b

ccan use unit vectors i,j

i vector length 1 in x direction

j vector length 1 in y direction

finding the angle between two vectors

2222cos

yxyx

yyxx

bbaa

baba

ab

ba

a,b, lengths of a,b

Result is a scalaryyxx babaabba cos

a

b

Vector producte.g. Find a vector perpendicular to two vectors

sinbac

bac

xyyx

zxxz

yzzy

zyx

zyx

baba

baba

baba

bbb

aaa

kji

bac

ˆˆˆ

a

b

c

Right-handed Co-ordinate system

i

j

r

y

x

r

position vary withˆ and ˆ of Directions r

Unit Vectors in Polar system

θ

r

x

y

r

i

j

r

jir ˆsinˆcosˆ

ji ˆcosˆsinˆ

The component of in the x direction = cosr cos

The component of in the y direction = sinr sin

r

r

The component of in the x direction = sinsinˆ

The component of in the y direction = coscosˆ

cosr

sinr cosˆ

sinˆ

θ

i

j

Velocity and acceleration vectors

• Position changes with time• Rate of change of r is

velocity– How much is the change in a

very small amount of time t

0 X

Y

x

r(t)r(t+t)t

trttr

dt

rdv

)()(

Limit at t0

(x,y) or (r,θ)

ˆˆ rrrv

Geometric interpretation of this equation

Radial component Tangential component

Relative Velocity 2D

V boat 2m/sV Alice 1m/s

V relative to shore

27,2/1tan

/521 22

smV

Relative Velocity 1De.g. Alice walks forwards along a boat at 1m/s and the boat moves at 2m/s. What is Alice’s velocity as seen by Bob ? If Bob is on the boat it is just 1 m/s If Bob is on the shore it is 1+2=3m/s If Bob is on a boat passing in the opposite direction….. and the earth is moving around the sun…

Velocity relative to an observer

e.g. Alice walks across the boat at 1m/s.As seen on the shore:

θ

Changing co-ordinate system

vt

Frame S (shore)

Frame S’ (boat) v boat w.r.t shore

(x’,y’)

Define the frame of reference – the co-ordinate system –in which you are measuring the relative motion.

No ‘correct’ or ‘preferred’ frame

x

x’

Equations for (stationary) Alice’s position on boat w.r.t shorei.e. the co-ordinate transformation from frame S to S’Assuming S and S’ coincide at t=0 :

Known as Gallilean transformationsThese simple relations do not hold in special relativity

y

• First Law– A body continues in a state of rest or uniform

motion unless there are forces acting on it.• No external force means no change in velocity

• Second Law– A net force F acting on a body of mass m [kg]

produces an acceleration a = F /m [ms-2]• Relates motion to its cause

F = ma units of F: kg.m.s-2, called Newtons [N]

Newton’s laws

We described the motion, position, velocity, acceleration,

now look at the underlying causes

• Third Law– The force exerted by A on B is equal and opposite to

the force exerted by B on A

Block on table

Weight

(a Force)

Fb

Fa

•Force exerted by block on table is Fa

•Force exerted by table on block is Fb

Fa=-Fb

(Both equal to weight)

Examples of Forces

weight of body from gravity (mg),

- remember m is the mass, mg is the force (weight)

tension, compression

friction, fluid resistance

Force Components

21 FFR

1F

2F

R

sin

cos

FF

FF

y

x

xF

yF F

iFF xxˆ

jFF yyˆ

•Force is a Vector•Resultant from vector sum

•Resolve into perpendicular components

Free Body Diagram• Apply Newton’s laws to particular body• Only forces acting on the body matter

– Net Force

• Separate problem into each body

Body 1

TensionIn rope

Block weightFriction

Body 2

Tension in rope

Block Weight

e.g.

F

Supporting Force from plane(normal force)

Tension & Compression• Tension

– Pulling force - flexible or rigid• String, rope, chain and bars

• Compression– Pushing force

• Bars

• Tension & compression act in BOTH directions.– Imagine string cut– Two equal & opposite forces – the tension

mgmg

mg

• A contact force resisting sliding– Origin is electrical forces between atoms in the two

surfaces.

• Static Friction (fs)

– Must be overcome before an objects starts to move

• Kinetic Friction (fk)

– The resisting force once sliding has started• does not depend on speed

Friction

mg

N

Ffs or fk

Nf

Nf

kk

ss

Friction – origin, values

• Friction proportional to N is an approximate rule

• Microscopic level– Intermolecular forces where surfaces come into contact– Once sliding starts usually easier to keep in motion

• Less bonding, kinetic < static friction

Material Static Coefficient Kinetic Coefficient

Steel on steelGlass on GlassTeflon on TeflonRubber on concrete (dry)Rubber on concrete (wet)

0.740.940.041.0

0.30

0.570.400.040.8

0.25

mg

Ff

x

yN

mg sin

mg cos

Show that when the block just begins to slide , μs = tanθ

Experimental determination of μs

The Rotor Fairground Ride

What does the speed of the Rotor need to be before the floor is removed?

Vehicles going round bendsCase A: Level Roads

Ff

Vehicles going round bendsCase A: Level Roads

mg

N

Ff

F

f

Vehicles going round bends

Case B – Banked roads

mg

N

N cos

N sin

mg

N

N

mg

Motion in a vertical circle

Looping the Loop

In a 1901 circus performance, Allo ‘Dare Devil’ Diavolo introduced the stunt of riding a bicycle in a loop the loop. Assuming that the loop is a circle of

radius R = 2.7m, what is the minimum speed Diavolo could have at the top of the loop in order to complete the stunt successfully?

Revision / Summary: Newton III • Newton III Pairs act on different bodies

– A on B, B on A– e.g. Earth pulls book down, book pulls Earth up

FTB

FEB

N.III Pairs: FTB = -FBT

FEB = -FBE

FBT

FBE

(aE=FBE/mE, tiny)

Revision / Summary: Newton II Problems1. Draw a simple sketch of the system to be analysed.2. Identify the individual objects to which Newton’s 2nd Law

can be applied.3. For each object draw a free-body diagram showing all the

forces acting on the object.4. Introduce a co-ordinate system for each object.5. For each object, determine the components of the forces

along each of the object’s co-ordinate axes.6. For each object, write a separate equation for each

component of Newton’s 2nd Law ( equation of motion).7. Solve the equations of motion.

Free BodyDiagram

TensionIn rope

Block weightFriction

(normal force)

backup

A passenger on a Ferris wheel moves in a vertical circle of radius R with constant speed, v. Assuming the seat remains upright during the motion, derive expressions for the force the seat exerts on the passenger at the top of the circle and at the bottom.

A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of 0.1 m. The hoop rotates at a constant rate of 4 revs/s about a vertical diameter.

(a) Find the angle β at which the bead is in vertical equilibrium.

(b) Is it possible for the bead to ‘ride’ at the same elevation as the centre of the hoop?

(c) What will happen if the hoop rotates at 1 rev/s ?