Presentation on theme: "Dynamics Dynamics Work/ Kinetic Energy Potential Energy"— Presentation transcript:
1 DynamicsDynamicsWork/ Kinetic Energy Potential EnergyConservative forcesConservation lawsMomentumCentre-of-massImpulseREAD theTextbook!Part II – “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”Chris ParkesOctober
2 Work is the change in energy that results from applying a force Work & EnergyWork is the change in energy that results from applying a forceWork = Force F times Distance s, units of Joules[J]More Precisely, W=F.xF,x Vectors so W=F x cosUnits (kg m s-2)m = Nm = J (units of energy)Note 1: Work can be negativee.g. Friction Force opposite direction to movement xNote 2: Can be multiple forces, uses resultant force ΣFNote 3: work is done on a specific body by a specific force (or forces)The rate of doing work is the Power [Js-1Watts]FsFxSo, for constant Force
3 in newtons acts on the particle. ExampleA particle is given a displacementin metres along a straight line. During the displacement, a constant forcein newtons acts on the particle.Find (a) the work done by the force and (b) the magnitude of the component of the force in the direction of the displacement.
5 Work Done by Varying Force 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 speedsuggestsWork Done by Varying ForceW=F.xbecomes
6 Energy, Work Potential Energy, U Energy can be converted into work Electrical, chemical, or letting aweight fall (gravitational)Hydro-electric power stationmgh of waterPotential Energy, UIn terms of the internal energy or potential energyPotential Energy - energy associated with the position or configuration of objects within a systemNote: Negative sign
7 Gravitational Potential Energy Choice of zero level is arbitrary Ug = mghmghReference planeUg = 0- hmgUg = - mghNo such thing as a definitive amount of PEmgparticle stays close to the Earth’s surface and so the gravitational force remains constant.
8 Stored energy in a Spring This stored energy has the potential to do work Potential EnergyWe are dealing with changes in energyhchoose 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 itHooke’s lawBUT, Force depends on extension xWork done by a variable force
9 Work done by a variable force Consider small distance dx over which force is constantF(x)Work W=Fx dxSo, total work is sumdxXFGraph of F vs x,integral is area under graphwork done = areadxX
10 Elastic Potential Energy Unstretched positionFor spring,F(x)=-kx:xFXX-XStretched spring stores P.E. ½kX2
11 Potential Energy Function kReference planexFsmg
12 Conservation of Energy K.E., P.E., Internal EnergyConservative & Dissipative ForcesConservative ForcesA 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 energythe amount of work done will depend on the path taken against the frictional forceOr fluid resistanceOr chemical energy of an explosion, adding mechanical energy
14 ExampleA 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.
15 Properties of conservative forces The work done by them is reversibleWork done on moving round a closed path is zeroThe work done by a conservative force is independent of the path, and depends only on the starting and finishing pointsBWork done by friction force is greater for this pathA
16 Forces and Energy e.g. spring Partial Derivative – derivative wrt one variable, others held constantGradient operator, said as grad(f)
17 Glider on a linear air track Negligible frictionMinimum on a potential energy curve is a position of stable equilibrium- no Force
18 Maximum on a potential energy curve is a position of unstable equilibrium
19 Linear Momentum Conservation Define momentum p=mvNewton’s 2nd law actuallySo, with no external forces, momentum is conserved.e.g. two body collision on frictionless surface in 1DAlso true for net forceson groups of particlesIfthenbeforem1m2v00 ms-1Initial momentum: m1 v0 = m1v1+ m2v2 : final momentumafterm1m2v2v1For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components
20 Energy measured in Joules [J] Energy ConservationEnergy can neither be created nor destroyedEnergy can be converted from one form to anotherNeed to consider all possible forms of energy in a system e.g:Kinetic energy (1/2 mv2)Potential energy (gravitational mgh, electrostatic)Electromagnetic energyWork done on the systemHeat (1st law of thermodynamics)Friction HeatEnergy measured in Joules [J]
21 Initial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E. Collision revisitedm1m2v2v1We identify two types of collisionsElastic: momentum and kinetic energy conservedInelastic: momentum is conserved, kinetic energy is notKinetic energy is transformed into other forms of energyInitial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E.m1>m2m1<m2m1=m2See lecture example for cases of elastic solutionNewton’s cradle
22 ImpulseChange in momentum from a force acting for a short amount of time (dt)NB: Just Newton 2nd law rewrittenWhere, p1 initial momentump2 final momentumQ) Estimate the impulseFor Andy Murray’s serve[135 mph]?ApproximatingderivativeImpulse is measured in Ns.change in momentum is measured in kg m/s.since a Newton is a kg m/s2 these are equivalent
23 Centre-of-mass Average location for the total mass Mass weighted average positionCentre of gravity – see textbookPosition vector of centre-of-mass
24 dm is mass of small element of body Rigid Bodies – Integral formydmrxzdm is mass of small element of bodyr is position vector of each small element.
25 Momentum and centre-of-mass Differentiating position to velocity:Hence momentum equivalent tototal mass × centre-of-mass velocityForces and centre-of-massDifferentiating velocity to acceleration:Centre-of-mass moves as acted on by the sum of the Forces acting
26 Internal Forces Internal forces between elements of the body and external forcesInternal forces are in action-reaction pairs and cancel in the sumHence only need to consider external forces on bodyIn terms of momentum of centre-of-mass
27 ExampleA 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-13ms-12kgC of M
28 Lab Frame before collision 4kg6 ms-13 ms-15 ms-12kgC of MCentre of Mass Frame before collision2 ms-11 ms-14kg2kgC of MCentre of Mass Frame after collision1 ms-12 ms-14kg2kgC of M