Dynamics READ the Textbook Work Kinetic Energy Potential

Dynamics READ the Textbook! Work/ Kinetic Energy Potential Energy Conservative forces Conservation laws Momentum Part II – “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. ” Centre-of-mass Impulse http: //www. hep. manchester. ac. uk/u/parkes/Chris_Parkes/Teaching. html October 2013 Chris Parkes

Work & Energy Work is the change in energy that results from applying a force • Work = Force F times Distance s, units of Joules[J] • More Precisely, W=F. x s – F, x Vectors so W=F x cos x F F – Units (kg m s-2)m = Nm = J (units of energy) – Note 1: Work can be negative • e. g. Friction Force opposite direction to movement x – Note 2: Can be multiple forces, uses resultant force ΣF – Note 3: work is done on a specific body by a specific force (or forces) • The rate of doing work is the So, for constant Force Power [Js-1 Watts]

Example A particle is given a displacement in metres along a straight line. During the displacement, a constant force in 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.

2 3 θ F cos θ F -4 -5 r

Work-Energy Theorem The 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. of particle is equal to the total work done to accelerate from rest to present speed suggests Work Done by Varying Force W=F. x becomes

Energy, Work • Energy can be converted into work – Electrical, chemical, or letting a weight fall (gravitational) mgh of water • Hydro-electric power station Potential Energy, U In terms of the internal energy or potential energy Potential Energy - energy associated with the position or configuration of objects within a system Note: Negative sign

Gravitational Potential Energy Choice of zero level is arbitrary Ug = mgh mg h Reference plane Ug = 0 mg -h Ug = - mgh mg No such thing as a definitive amount of PE particle stays close to the Earth’s surface and so the gravitational force remains constant.

This stored energy has the potential to do work Potential We are dealing with changes in energy Energy h • choose an arbitrary 0, and look at p. e. 0 This was gravitational p. e. , another example : Stored energy in a Spring Do work on a spring to compress it or expand it Hooke’s law BUT, Force depends on extension x Work done by a variable force

Work done by a variable force Consider small distance dx over which force is constant F(x) dx Work W=Fx dx So, total work is sum X 0 Graph of F vs x, F integral is area under graph work done = area dx X

Elastic Potential Energy Unstretched position For spring, F(x)=-kx: x F X X -X Stretched spring stores P. E. ½k. X 2

Potential Energy Function k Reference plane x Fs mg

Conservation of Energy K. E. , P. E. , Internal Energy Conservative & Dissipative Forces • Conservative Forces – A 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 energy – the amount of work done will depend on the path taken against the frictional force – Or fluid resistance – Or chemical energy of an explosion, adding mechanical energy

Conservative forces frictionless surface

Example A 2 kg 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 = 1 m, if it is released from rest at A.

Properties of conservative forces • The work done by them is reversible • Work done on moving round a closed path is zero • The work done by a conservative force is independent of the path, and depends only on the starting and finishing points B Work done by friction force is greater for this path A

Forces and Energy e. g. spring • Partial Derivative – derivative wrt one variable, others held constant • Gradient operator, said as grad(f)

Glider on a linear air track Negligible friction Minimum on a potential energy curve is a position of stable equilibrium - no Force

Maximum on a potential energy curve is a position of unstable equilibrium U

Linear Momentum Conservation • Define momentum p=mv • Newton’s 2 nd law actually • So, with no external forces, momentum is conserved. Also true for net forces on groups of particles • e. g. two body collision on frictionless If surface in 1 D then before m 1 m 2 0 ms-1 Initial momentum: m 1 v 0 = m 1 v 1+ m 2 v 2 : final momentum after v 0 m 1 m 2 v 1 For 2 D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components

Energy Conservation • Energy can neither be created nor destroyed • Energy can be converted from one form to another • Need to consider all possible forms of energy in a system e. g: – – – Kinetic energy (1/2 mv 2) Potential energy (gravitational mgh, electrostatic) Electromagnetic energy Work done on the system Heat (1 st law of thermodynamics) • Friction Heat Energy measured in Joules [J]

Collision revisited m 1 v 1 m 2 • We identify two types of collisions – Elastic: momentum and kinetic energy conserved Initial K. E. : ½m 1 v 02 = ½ m 1 v 12+ ½ m 2 v 22 : final K. E. – Inelastic: momentum is conserved, kinetic energy is not • Kinetic energy is transformed into other forms of energy See lecture example for cases of elastic solution Newton’s cradle 1. m 1>m 2 2. m 1<m 2 3. m 1=m 2 v 2

Impulse • Change in momentum from a force acting for a short amount of time (dt) Where, p 1 initial momentum p 2 final momentum • NB: Just Newton 2 nd law rewritten Approximating derivative Impulse is measured in Ns. change in momentum is measured in kg m/s. since a Newton is a kg m/s 2 these are equivalent Q) Estimate the impulse For Andy Murray’s serve [135 mph]?

Centre-of-mass • Average location for the total mass Mass weighted average position Centre of gravity – see textbook • Position vector of centre-of-mass

y Rigid Bodies – Integral form r dm x z dm is mass of small element of body r is position vector of each small element.

Momentum and centre-of-mass • Differentiating position to velocity: • Hence momentum equivalent to total mass × centre-of-mass velocity Forces and centre-of-mass • Differentiating velocity to acceleration: • Centre-of-mass moves as acted on by the sum of the Forces acting

Internal Forces • Internal forces between elements of the body • and external forces – Internal forces are in action-reaction pairs and cancel in the sum – Hence only need to consider external forces on body • In terms of momentum of centre-of-mass

Example • A body moving to the right collides elastically with a 2 kg body moving in the same direction at 3 m/s. The collision is head-on. Determine the final velocities of each body, using the centre of mass frame. 6 ms-1 4 kg C of M 2 kg 3 ms-1

Lab Frame before collision 6 ms-1 4 kg 5 ms-1 2 kg C of M 3 ms-1 Centre of Mass Frame before collision 4 kg 1 ms-1 C of M 2 ms-1 2 kg Centre of Mass Frame after collision 1 ms-1 4 kg C of M 2 kg 2 ms-1