Work Work – a transfer of energy from one object to another by mechanical means mechanical - something physically makes the transfer (as opposed to a wave, which is pure energy) Ex: If you lift a book above your head, you did work on the book you lost some energy, the book gains that energy, by being higher off the ground

The Math of Work Eq’n: Work = Force • displacement W = FΔx Units: Joules = Newton•meter J = kg m/s2 • m J = kg m2/s2 where Δx measurement is the straight line from start to finish, not path dependent so how much work done if you run in a circle? so how much work done if you support a book out at your side? where the F and the Δx it’s moved are assumed to be parallel to each other…

So if the force and displacement are parallel to each other, you’d use the entire value of F in the equation, But if they are at an angle to each other, which is far more likely in the real world, then you need to find the component of the force that’s parallel to the displacement, and only the value of the component is used as F in the equation.

Work is actually a scalar quantity so + or – does not indicate direction! positive W Refers to work that’s done on an object, so that object gains energy When F (or its contributing component) & Δx are in the same direction negative W Refers to work that’s done by an object, so that object loses energy When F (or its contributing component) & Δx are in the opposite directions Ex: work done by friction or gravity, if going up

The Energy Outline Energy - the ability to do work or the ability to change an object or its surroundings I. Mechanical NRG – energy due to a physical situation A. Kinetic NRG (KE) – energy of motion an object must have speed Eq’ns: KE = ½ mv2 and ΔKE = ½ m(vf2 - vi2) units: J = kgm2/s2 (recall from work: J = Nm = kgm2/s2)

B. Potential NRG (PE) – energy of position, stored away, ready to set an object into motion 1. Gravitational PE (PEg) – object’s position is some height different than the reference level, where PEg = 0 Eq’ns: PEg = mgh = Fgh & ΔPEg = mg(hf – hi)

2. Elastic PE – object’s position is some shape different than normal

3. Chemical PE – object’s atoms/molecules have the potential to rearrange 4. Electric PE – the charged particles within the object have the potential to move – create electricity!

II. Thermal NRG – energy of heat – the energy that is produced when it looks like energy isn’t being conserved… Examples of ways to create it: friction air resistance sound deformation

The Work – Energy Theorem This is the connection between work and energy: The amount of work done by* / on** an object will equal its change in energy * “by” – object will lose NRG, so -ΔE ** “on” – object will gain NRG, so +ΔE In effect, work is the act of changing an object’s NRG, while energy means work can get done. PE means it has the potential to get done. KE means it is presently getting done. Eq’n: W = ΔE, or more specifically, W = ΔKE + ΔPE

Back to KE – what does it mean to have velocity2 in the equation? Let’s consider how long it takes to stop a vehicle traveling at a particular speed: At first you may think: if you’re traveling twice as fast, it will take twice as far to stop, But that’s not right… W = ΔKE or F Δx = ½ m Δv2 So as v increases, both KE and the other side of the equation goes up by the square of that increase, Ex: for 2x’s as much v, there’s 4x’s as much KE in the vehicle, there’s 4x’s as much W needed to stop it, and for a constant braking force, that means 4x’s as much distance is required!

Conservation of Energy The Law of Conservation of Energy: NRG can’t be created or destroyed, but it can be changed from one form to another. Eq’n: Ei = Ef Possible forms these NRG’s could be in? in the ideal world: only ME (KE & PE) in the real world: ME (KE & PE) & TE Consider the double incline ramp: Will the ball roll off the short side if released from the long side? No, but why? Now we can answer that…

The double incline ramp with 4 noted positions: 1: E1 = PEg1 Since there’s h, but no v 2: E2 = PEg2 + KE2 Since still some h, but also v 3: E3 = KE3 Since no more h*, and all v 4: E4 = PEg4 Since no v, and back at original h – but is it? *If RL defined as top of metal track at bottom of ramp L of C of E says: E1 = E2 = E3 = E4 So in the ideal world: PEg1 = PEg2 + KE2 = KE3 = PEg4 so mgh1 = mgh4 then h1 = h4 But in the real world: PEg1 = PEg2 + KE2 + TE2 = KE3 + TE3= PEg4 + TE4 so mgh1 = mgh4 + TE4 then h1 > h4

Swinging Pendulum on a string 2 forms of ME that the total amount of NRG fluctuates between max KE at equilibrium (vertical) position max PEg at either end if real world: loss of ME to friction where string rubs on support & air resistance Bobbing Mass on a spring 3 forms of ME that the total amount of NRG fluctuates between max KE at the equilibrium (midpoint) position max PEe at the bottom and some at the top max PEg at the top if real world: loss of ME to friction inside spring & air resistance

Mutli-Color Tracks Which shape track wins? One with its dip 1st, wins – gets to a faster speed earlier in its trip Which shape track gets the ball going the fastest by the end? Ideal: All have = PEg at start, so = KE at end, so = v too Real: the shortest distance (straight) track loses least KE to TE, so it’s going a little faster

The Math of Conservation of Energy ID givens, unknown & reference level! Can draw a diagram of “i” & “f” situations to help 2. state: MEi = MEf (we assume ideal) 3. then: PEgi + KEi = PEgf + KEf only of needed terms 4. then: mghi + ½mvi2 = mghf + ½mvf2 where mass will often cancel watch units: need kg, m, s  Joules Ex: A 0.4 kg ball is dropped 2m. What’s its speed when it reaches the ground?

The Math of Conservation of Energy ID givens & unknown Draw a diagram of “i” & “f” situations use “fast marks” to indicate speed ID RL to indicate height 3. state: MEi = MEf (we assume ideal) 4. then: PEgi + KEi = PEgf + KEf so are any of the terms = 0? 5. then: mghi + ½mvi2 = mghf + ½mvf2 where mass will often cancel watch units (kg, m, s  Joules) Ex: A 0.4 kg ball is dropped 2m. What’s its speed when it reaches the ground?

Consider a story about 2 students asked to help get books loaded into the cabinets. Both move equal numbers of books up to equal height cabinets, but one does it quickly and efficiently, while the other does not. Who does more work? Neither! Both apply same F thru = Δx Who uses more power? The quicker one Power is the rate at which work gets done Eq’n: P = W/Δt units: Watts = J/s (1 hp = 746 Watts)