Honors Physics, Pg 1 Physics II Today’s Agenda l Work & Energy. çDiscussion. çDefinition. l Work of a constant force. l Power l Work kinetic-energy theorem. l Work of a sum of constant forces. l Work for a sum of displacements with constant force. l Work done by a spring l Conservation of Energy l Comments.

Honors Physics, Pg 2 Work & Energy l One of the most important concepts in physics. çAlternative approach to mechanics. l Many applications beyond mechanics. çThermodynamics (movement of heat). çQuantum mechanics... l Very useful tools. çYou will learn new (sometimes much easier) ways to solve problems. See text

Honors Physics, Pg 3 Forms of Energy l Kinetic l Kinetic: Energy of motion. çA car on the highway has kinetic energy. çWe have to remove this energy to stop it. çThe breaks of a car get HOT ! çThis is an example of turning one form of energy into another. (More about this soon)...

Honors Physics, Pg 4 Forms of Energy l Potential l Potential: Stored, “potentially” ready to use. çGravitational. »Hydro-electric dams etc... çElectromagnetic »Atomic (springs, chemical...) çNuclear »Sun, power stations, bombs...

Honors Physics, Pg 5 Mass = Energy l Particle Physics: + 5,000,000,000 V e- - 5,000,000,000 V e+ (a) (b) (c) E = eV M E = MC 2 ( poof ! )

Honors Physics, Pg 6 Energy Conservation l Energy cannot be destroyed or created. çJust changed from one form to another. energy is conserved l We say energy is conserved ! çTrue for any isolated system. çi.e when we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-breaks-road-atmosphere” system is the same. çThe energy of the car “alone” is not conserved... »It is reduced by the braking. workenergy l Doing “work” on a system will change it’s “energy”...

Honors Physics, Pg 7 Definition of Work: Ingredients: FS Ingredients: Force ( F ), displacement ( S ) F Work, W, of a constant force F S acting through a displacement S is: F. S W = F. S = FScos(  ) = F S S  F S displacement FSFS “Dot Product”

Honors Physics, Pg 8 Work: 1-D Example (constant force) F x A force F = 10N pushes a box across a frictionless floor for a distance  x = 5m. xxxx F byF on FxFx Work done by F on box : W F = F.  x = F  x (since F is parallel to  x) W F = (10 N)x(5m) = 50N-m. See example 7.1

Honors Physics, Pg 9 Units: N-m (Joule) Dyne-cm (erg) = J BTU= 1054 J calorie= J foot-lb= J eV= 1.6x J cgsothermks Force x Distance = Work Newton x [M][L] / [T] 2 Meter = Joule [L] [M][L] 2 / [T] 2

Honors Physics, Pg 10 Power FS l We have seen that W = F.  S çThis does not depend on time ! l Power is the “rate of doing work”: FSFv Fv l If the force does not depend on time:  W/  t = F.  S/  t = F. v P = F. v l Units of power: J/sec = Nm/sec = WattsFS v

Honors Physics, Pg 11 Comments: l Time interval not relevant. çRun up the stairs quickly or slowly...same W. FS Since W = F. S l No work is done if: çF çF = 0 or çS çS = 0 or   = 90 o

Honors Physics, Pg 12 Comments... FS W = F. S No work done if  = 90 o. T çNo work done by T. çNo work done by N. T v v N

Honors Physics, Pg 13 Work & Kinetic Energy: F x A force F = 10N pushes a box across a frictionless floor for a distance  x = 5m. The speed of the box is v 1 before the push, and v 2 after the push. xxxx F v1v1 v2v2i m

Honors Physics, Pg 14 Work & Kinetic Energy... Fa l Since the force F is constant, acceleration a will be constant. We have shown that for constant a:  v v 1 2 = 2a(x 2 -x 1 ) = 2a  x.  multiply by 1 / 2 m: 1 / 2 mv / 2 mv 1 2 = ma  x  But F = ma 1 / 2 mv / 2 mv 1 2 = F  x xxxx F v1v1 v2v2a i m

Honors Physics, Pg 15 Work & Kinetic Energy... l So we find that  1 / 2 mv / 2 mv 1 2 = F  x = W F l Define Kinetic Energy K:K = 1 / 2 mv 2 çK 2 - K 1 = W F (Work kinetic-energy theorem)  W F =  K (Work kinetic-energy theorem) xxxx F v1v1 v2v2a i m

Honors Physics, Pg 16 Work Kinetic-Energy Theorem: NetWork {Net Work done on object} = changekinetic energy {change in kinetic energy of object} l This is true in general: K1K1 K2K2 F F netdS

Honors Physics, Pg 17 Work done by Variable Force: (1D) When the force was constant, we wrote W = F  x çarea under F vs x plot: l For variable force, we find the area by integrating: çdW = F(x) dx. F x WgWg xx F(x) x1x1 x2x2 dx

Honors Physics, Pg 18 A simple application: Work done by gravity on a falling object l What is the speed of an object after falling a distance H, assuming it starts at rest ? FS l W g = F. S = mgScos(0) = mgH W g = mgH Work Kinetic-Energy Theorem: W g = mgH = 1 / 2 mv 2 S gmggmg H j v 0 = 0 v

Honors Physics, Pg 19 Conservation of Energy l If only conservative forces are present, the total energy (sum of potential and kinetic energies) of a system is conserved (i.e. constant). constant!!! l If only conservative forces are present, the total energy (sum of potential and kinetic energies) of a system is conserved (i.e. constant). E = K + U is constant !!! l Both K and U can change as long as E = K + U is constant.

Honors Physics, Pg 20 Example: The simple pendulum. l Suppose we release a bob or mass m from rest a distance h 1 above it’s lowest possible point. çWhat is the maximum speed of the bob and where does this happen ? çTo what height h 2 does it rise on the other side ? v h1h1 h2h2 m See example A Pendulum

Honors Physics, Pg 21 Example: The simple pendulum. l Energy is conserved since gravity is a conservative force (E = K + U is constant) l Choose y = 0 at the original position of the bob, and U = 0 at y = 0 (arbitrary choice). E = 1 / 2 mv 2 + mgy. v h1h1 h2h2 y y=0 See example, A Pendulum

Honors Physics, Pg 22 Example: The simple pendulum. l E = 1 / 2 mv 2 + mgy. çInitially, y = 0 and v = 0, so E = 0. çSince E = 0 initially, E = 0 always since energy is conserved. y y=0 See example, A Pendulum

Honors Physics, Pg 23 Example: The simple pendulum. l E = 1 / 2 mv 2 + mgy. l So at y = -h, E = 1 / 2 mv 2 - mgh = 0. 1 / 2 mv 2 = mgh l 1 / 2 mv 2 will be maximum when mgh is minimum. l 1 / 2 mv 2 will be maximum at the bottom of the swing ! y y=0 y=-h h See example, A Pendulum

Honors Physics, Pg 24 Example: The simple pendulum. l 1 / 2 mv 2 will be maximum at the bottom of the swing ! l So at y = -h 1 1 / 2 mv 2 = mgh 1 v 2 = 2gh 1 v h1h1 y y=0 y=-h 1 See example, A Pendulum

Honors Physics, Pg 25 Example: The simple pendulum. l Since 1 / 2 mv 2 - mgh = 0 it is clear that the maximum height on the other side will be at y = 0 and v = 0. l The ball returns to it’s original height. y y=0 See example, A Pendulum

Honors Physics, Pg 26 Example: The simple pendulum. l The ball will oscillate back and forth. The limits on it’s height and speed are a consequence of the sharing of energy between K and U. E = 1 / 2 mv 2 + mgy = K + U = 0. y See example A Pendulum

Honors Physics, Pg 27 Vertical Springs and HOOKE’S LAW l A spring is hung vertically, it’s relaxed position at y=0 (a). When a mass m is hung from it’s end, the new equilibrium position is y E (b). l Hook’s Law relates the force exerted by the spring with the elongation of the spring l Force exerted by the spring is directly proportional to it’s elongation from it’s resting position l F=-kx(negative sign shows that the force is in the opposite direction of the force) l F=mg when spring is elongated and nonmoving so that mg=kx x = 0 X=x f j k m (a) (b) mg

Honors Physics, Pg 28 Vertical Springs l If we choose x = 0 to be at the equilibrium position of the mass hanging on the spring, we can define the potential in the simple form. l Notice that g does not appear in this expression !! çBy choosing our coordinates and constants cleverly, we can hide the effects of gravity. x = 0 j k m (a) (b)

Honors Physics, Pg 29 1-D Variable Force Example: Spring l For a spring we know that F x = -kx. F(x) x2x2 x x1x1 -kx equilibrium F = - k x 1 F = - k x 2

Honors Physics, Pg 30 Spring... l The work done by the spring W s during a displacement from x 1 to x 2 is the area under the F(x) vs x plot between x 1 and x 2. WsWs F(x) x2x2 x x1x1 -kx equilibrium

Honors Physics, Pg 31 Spring... F(x) x2x2 WsWs x x1x1 -kx l The work done by the spring W s during a displacement from x 1 to x 2 is the area under the F(x) vs x plot between x 1 and x 2.

Honors Physics, Pg 32 Non-conservative Forces: l If the work done does not depend on the path taken, the force involved is said to be conservative. l If the work done does depend on the path taken, the force involved is said to be non-conservative. l An example of a non-conservative force is friction: l Pushing a box across the floor, the amount of work that is done by friction depends on the path taken. çWork done is proportional to the length of the path !

Honors Physics, Pg 33 Non-conservative Forces: Friction Suppose you are pushing a box across a flat floor. The mass of the box is m and the kinetic coefficient of friction is . F. D The work done in pushing it a distance D is given by: W f = F f. D = -  mgD. D F f = -  mg

Honors Physics, Pg 34 Non-conservative Forces: Friction Since the force is constant in magnitude, and opposite in direction to the displacement, the work done in pushing the box through an arbitrary path of length L is just W f = -  mgL. l Clearly, the work done depends on the path taken. l W path 2 > W path 1. A B path 1 path 2 See text: 8-6

Honors Physics, Pg 35 Generalized Work Energy Theorem: l Suppose F NET = F C + F NC (sum of conservative and non- conservative forces). l The total work done is: W TOT = W C + W NC The Work Kinetic-Energy theorem says that: W TOT =  K.  W TOT = W C + W NC =  K But W C = -  U So W NC =  K +  U =  E or W NC = E i - E f

Honors Physics, Pg 36 Generalized Work Energy Theorem: l The change in total energy of a system is equal to the work done on it by non-conservative forces. E of system not conserved ! l Or the Potential Energy + Kinetic Energy + Internal Energy is a constant equal to the Total Energy  If all the forces are conservative, we know that energy is conserved:  K +  U =  E = 0 which says that W NC = 0, which makes sense. çIf some non-conservative force (like friction) does work, energy will not be conserved by an amount equal to this work, which also makes sense. W NC =  K +  U =  E