Dynamics Velocity Acceleration Inertial Frames Forces Newtons Laws

Dynamics Velocity & Acceleration Inertial Frames Forces – Newton’s Laws 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! http: //www. hep. manchester. ac. uk/u/parkes/Chris_Parkes/Teaching. html October 2013 Chris Parkes

vector addition • c=a+b y cx= ax +bx cy= ay +by can use unit vectors i, j b a c i vector length 1 in x direction x j vector length 1 in y direction scalar product finding the angle between two vectors a, b, lengths of a, b Result is a scalar a b

Vector product e. g. Find a vector perpendicular to two vectors c Right-handed Co-ordinate system b a

Unit Vectors in Polar system y r θ x

y θ x The component of in the x direction = in the y direction = The component of in the x direction = The component of in the y direction =

Velocity and acceleration vectors • Position changes with time • Rate of change of r is velocity Y (x, y) or (r, θ) – How much is the change in a very small amount of time t r(t) Limit at t 0 0 x r(t+ t) X

Geometric interpretation of this equation Radial component Tangential component

Relative Velocity 1 D e. g. Alice walks forwards along a boat at 1 m/s and the boat moves at 2 m/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=3 m/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 Relative Velocity 2 D e. g. Alice walks across the boat at 1 m/s. V boat 2 m/s As seen on the shore: θ V relative to shore V Alice 1 m/s

Changing co-ordinate system y Define the frame of reference – the co-ordinate system – in which you are measuring the relative motion. No ‘correct’ or ‘preferred’ frame (x’, y’) Frame S’ (boat) v boat w. r. t shore Frame S (shore) vt x’ x Equations for (stationary) Alice’s position on boat w. r. t shore i. e. the co-ordinate transformation from frame S to S’ Assuming S and S’ coincide at t=0 : Known as Gallilean transformations These simple relations do not hold in special relativity

We described the motion, position, velocity, acceleration, now look at the underlying causes Newton’s laws • 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]

• Third Law – The force exerted by A on B is equal and opposite to the force exerted by B on A Fb • Force exerted by Block on table block on table is F Fa=-Fb Weight (a Force) Fa • Force exerted by table on block is 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 a

Force Components • 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 e. g. Body 1 Supporting Force from plane (normal force) Friction Tension In rope Block weight Body 2 Tension in rope Block Weight

Tension & Compression • Tension – Pulling force - flexible or rigid • String, rope, chain and bars mg • Compression – Pushing force • Bars mg mg • Tension & compression act in BOTH directions. – Imagine string cut – Two equal & opposite forces – the tension

Friction • 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 N fs or fk F mg

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 Steel on steel Glass on Glass Teflon on Teflon Rubber on concrete (dry) Rubber on concrete (wet) Static Coefficient 0. 74 0. 94 0. 04 1. 0 0. 30 Kinetic Coefficient 0. 57 0. 40 0. 04 0. 8 0. 25

Experimental determination of μs N y Ff mg cos mg mg sin x Show that when the block just begins to slide , μs = tanθ

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

Vehicles going round bends Case A: Level Roads Ff

Vehicles going round bends Case A: Level Roads N Ff mg F f

Vehicles going round bends Case B – Banked roads N N cos N sin mg

Motion in a vertical circle N N mg mg

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. 7 m, 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 FBT FEB FBE N. III Pairs: FTB = -FBT FEB = -FBE (a. E=FBE/m. E, tiny)

Revision / Summary: Newton II Problems 1. 2. 3. 4. 5. 6. 7. Draw a simple sketch of the system to be analysed. Identify the individual objects to which Newton’s 2 nd Law can be applied. For each object draw a free-body diagram showing all the forces acting on the object. Introduce a co-ordinate system for each object. For each object, determine the components of the forces along each of the object’s co-ordinate axes. For each object, write a separate equation for each component of Newton’s 2 nd Law ( equation of motion). Solve the equations of motion. Tension (normal In rope Free Body force) Diagram Friction Block weight

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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 ?