Dynamics Chris Parkes ctober 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!
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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 ?