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EE1 Particle Kinematics : Newton’s Legacy "If I see further, it is because I stand on the shoulders of giants," Chris Parkes October 2005 Motion Forces Energy & Momentum Conservation Circular Motion Gravity

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Motion Position [m] Velocity [ms -1 ] –Rate of change of position Acceleration [ms -2 ] –Rate of change of velocity t x v t dx dt e.g 0 a 0 0

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Equations of motion in 1D –Initially (t=0) at x 0 –Initial velocity u, –acceleration a, s=ut+1/2 at 2, where s is displacement from initial position v=u+at Differentiate w.r.t. time: v 2 =u 2 +2 as

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2D motion: vector quantities Position is a vector –r, (x,y) or (r, ) –Cartesian or cylindrical polar co- ordinates –For 3D would specify z also Right angle triangle x=r cos , y=r sin r 2 =x 2 +y 2, tan = y/x Scalar: 1 number Vector: magnitude & direction, >1 number 0 X Y x y r

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vector addition c=a+b c x = a x +b x c y = a y +b y scalar product x y a b c can 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 a,b, lengths of a,b Result is a scalar a b

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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) Limit at t 0

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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

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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) FbFb FaFa Force exerted by block on table is F a Force exerted by table on block is F b F a =-F b (Both equal to weight) Examples of Forces For this course: weight of body from gravity (mg), tension, compression Friction,

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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 mg

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A contact force resisting sliding –Origin is chemical forces between atoms in the two surfaces. Static Friction (f s ) –Must be overcome before an objects starts to move Kinetic Friction (f k ) –The resisting force once sliding has started does not depend on speed Friction mg N F f s or f k

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Questions Topics Covered –Vector addition and dot product, descriptions of motion, Newton’s 3 laws, Friction. ChapterPageExerciseTopic 2279Vector addition 35012Motion 35134Motion 59619Newton’s 2nd law ChapterPageExerciseTopic 22826Vector addition 22946Vector dot product 35020Motion 35242Motion 35362Motion 59625Newton’s 2 nd law 61194Friction Friction 61243Newton’s laws From Benson, University Physics, Revised Addition

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