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

Work is the change in energy that results from applying a force 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 F,x Vectors so W=F x cos 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 Power [Js-1Watts] F s F  x So, for constant Force

in newtons acts on the particle. 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 r - 5

Work Done by Varying Force 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. 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 Potential Energy, U Energy can be converted into work Electrical, chemical, or letting a weight fall (gravitational) Hydro-electric power station mgh of water 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 - h mg Ug = - mgh No such thing as a definitive amount of PE mg particle stays close to the Earth’s surface and so the gravitational force remains constant.

Stored energy in a Spring This stored energy has the potential to do work Potential Energy We are dealing with changes in energy h choose an arbitrary 0, and look at  p.e. 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) Work W=Fx dx So, total work is sum dx X F Graph of F vs x, 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. ½kX2

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 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.

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

Linear Momentum Conservation Define momentum p=mv Newton’s 2nd law actually So, with no external forces, momentum is conserved. e.g. two body collision on frictionless surface in 1D Also true for net forces on groups of particles If then before m1 m2 v0 0 ms-1 Initial momentum: m1 v0 = m1v1+ m2v2 : final momentum after m1 m2 v2 v1 For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components

Energy measured in Joules [J] 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 mv2) Potential energy (gravitational mgh, electrostatic) Electromagnetic energy Work done on the system Heat (1st law of thermodynamics) Friction  Heat Energy measured in Joules [J]

Initial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E. Collision revisited m1 m2 v2 v1 We identify two types of collisions Elastic: momentum and kinetic energy conserved Inelastic: momentum is conserved, kinetic energy is not Kinetic energy is transformed into other forms of energy Initial K.E.: ½m1 v02 = ½ m1v12+ ½ m2v22 : final K.E. m1>m2 m1<m2 m1=m2 See lecture example for cases of elastic solution Newton’s cradle

Impulse Change in momentum from a force acting for a short amount of time (dt) NB: Just Newton 2nd law rewritten Where, p1 initial momentum p2 final momentum Q) Estimate the impulse For Andy Murray’s serve [135 mph]? Approximating derivative Impulse is measured in Ns. change in momentum is measured in kg m/s. since a Newton is a kg m/s2 these are equivalent

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

dm is mass of small element of body Rigid Bodies – Integral form y dm r 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 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. 4kg 6ms-1 3ms-1 2kg C of M

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