Work

1. Work what’s accomplished by a force when it makes an object move through a displacement OR a transfer of energy from one object to another by mechanical means mechanical - something physically makes the transfer (As opposed to transferring energy by means of a wave…) 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

2. The Math of Work Eq’n: Work = F•Δd Note, this is a Dot Product of two vectors – see next slide. or W = FΔd cos where is the angle between their tails (Text uses W = FllΔd – Yuck) Units: Joules = Newton∙meter J = kg m/s2 ∙ m J = kg m2/s2 also 1 J = 107 erg = 107 dyne cm = .738 ftlb Note Δd (or Δx,Δh, ∆y,l) is displacement, so not path dependent. How much work done if you run in a circle? How much work done if you support a book out at your side?

3. Recall math computations with vectors from Ch 3: V times V has 2 possible results, depending on how it’s defined: Dot Product – defines the multiplication of the vectors in terms of the components of them that are parallel to each other, so the answer ends up a scalar again. The Dot Product expression tells the user to actually find the product of just the magnitudes of the two vectors involved, but then also multiply by the cosine of the angle between those two vectors’ tails to ensure you’re only using the parallel components of the two vectors to get you’re final answer. Cross Product – defines the multiplication of the vectors in terms of the components of them that are perpendicular to each other, so that the answer ends up another vector, perpendicular in direction to either of the original vectors – we’ll need in Ch 8.

4. Then work is actually a scalar quantity so + or – does not indicate direction! positive W when F& Δd (or components) in same direction we’ll see this is actually a gain in energy negative W when F & Δd (or comp) in opposite directions we’ll see this is actually a loss of energy Ex: work done by friction or work done by gravity, if object is going up

5. So then what force is responsible for work done when it seems like F is  to Δd? Consider a bag with a strap: There is an applied force against the tiny piece of a strap that’s against the front face of your finger to get it going, but then that’s it – nothing needs to keep it going… Newton’s 1st law… Consider a waitress tray carried flat on your hand: There is a force of static friction between the tray and your hand that gets it going, and again, that’s all you need… So in either case, there can be work being done, just not by the most obvious force. Consider an object moved around a curve by a Fc: Since Fc is always  to v & Δx, it does no work!

6. For these reasons, when dealing with work it’s important to specify which force is responsible for the work getting done, so all uses of W = F Δd cosmust have subscripts on W & F that match! Let’s do a few together: Ex 1. A person pulls a 50 kg crate 40 m with a force of 100 N at 37 with the floor, where it runs into 50 N of friction. Determine a) the work done by each of the forces acting and b) the net work done on the crate. Ex 2. Determine the work a 65 kg student does a) on a 15 kg backpack that he carries at a steady pace up the steps of a stadium, which are 20 m long and at a 30 angle with the ground. b) Work to carry himself to top?

7. Graphing Work 1st, the area under the curve of a F vs Δx graph = Work Since area = bh, or ½bh, or area of a trapoziod… 2nd, the general definition of work done by a force must take into account the fact that the force may vary in both mag & direction, and that the path followed may also change in direction. All these things can be taken into account by defining work as an integral…

8. Work Done by a Varying Force

9. Power 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 same Δd Who uses more power? The quicker one Power is the rate at which work gets done or the rate at which energy is transformed Eq’n: P = W/Δt = ΔE/Δt or P = FΔdcos/Δt = Fvavg units: Watts = J/s also 1 horsepower = 550 ft lb/s = 746 Watts

10. Efficiency an important characteristic of all machines how much do you have to put into them, Pin vs how much you can get out of them, Pout eff = Pout/Pin x 100% always less than 100%! typical internal combustion engine (car): 15% standard incandescent light bulb: 10% CFL bulbs: 40% (plus last 8-15x’s longer!) human body: 25%

11. 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 (K) – energy of motion an object must have speed Eq’ns for translational* K: K = ½ mv2 & ΔK = ½ m(vf2 - vi2) units: J = kg m2/s2 (recall from work: J = Nm = kg m2/s2) *as opposed to rotational K…Ch 8

12. B. Potential NRG (U) – energy of position, associated with forces that depend on the configuration of an object relative to its surroundings 1. Gravitational PE (Ug) – from Fg – object’s position is some vertical height different than the reference level, defined as where Ug = 0 Eq’ns: Ug= mgh = Fgh & ΔUg = mg(hf– hi) or, for celestial objects… “-” since UGdefined as = 0 at infinity…

13. 2. Elastic PE (Us) – from Fs – object’s position is some shape different than normal Eq’ns (for Hooke’s law apparatus – explained next): Us = ½kx2 & ΔUs = ½k(xf2 - xi2) In Physicsland, just about anything that stretches or compresses is assumed to be follow Hooke’s law…

14. Now, an aside from NRG to understand Hooke’s law… In the ideal case, elastic PE is most commonly from a Hooke’s law apparatus… which is any object that’s deformation has a linear relationship with the force the object applies to restore itself to its original shape when deformed traditionally expressed as Fs x, where Fs is the force the spring, or any deformed object, applies when distorted from original shape and x is deformation of object (Not original length/size!) and the constant of proportionality is easily determined experimentally…

15. If we set up the lab as follows: use a spring scale to extend a spring a predetermined amount of deformation, then which is ind? __ & which dep? __ Which means your graph should be of __ vs __ & the constant of proportionality (slope) is known as the spring constant, “k” tells how strong/weak spring is in units of N/m So the spring eq’n or Hooke’s law is: Fs=kx or Fs=-kx shown with “–” when not written as vectors since F, the restoring force of the spring, will always act in opposite direction as x, the deformation.

16. But there’s more! Recall in our study of work, the area under the graph = the work done… But graphing is only necessary when the force isn’t constant… like when deforming an elastic object! so the graph of F vs x for a Hooke’s law apparatus can tell us the same thing! Area = Work done by the outside force that’s deformed the object, by W = area = ½bh where b=x & h = F = kx (from spring eq’n) so W = ½ x kx = ½ kx2 … look familiar? So area = amount of Us stored in object too! Pretty cool, ehh??

17. Internal NRG – sum total of all the NRG of all the molecules in an object unable to be observed by looking at an object what’s produced when it looks like energy isn’t being conserved… A. Kinetic – aka Thermal (Q) – Always created in real life (recall from Chem: Q = 3/2nRT??) 1. friction (& air resistance) 2. sound 3. deformation permanent temporary – internal friction during change

18. B. Potential – we won’t deal with these in Ch6 math… 1. Chemical PE – object’s atoms/molecules have the potential to rearrange 2. Electric PE – the charged particles (e-) within the object have the potential to move – create electricity!

19. 3. Nuclear PE – subatomic particles (p+, n0) have potential to move out of/into nucleus of the atom Fission is when a nucleus splits Fusion is when 2 nuclei are joined together Cold fusion is considered to be one of the “holy grails” of physics research… In either case, mass is converted into NRG by E = mc2… so a little m makes A LOT of E!

20. A Diagram of a Nuclear Reactor

21. 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 this energy could be in? in the ideal world: only ME in the real world: ME & IE 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…

22. The double incline ramp with 4 noted positions: 1: E1 = Ug1 Since there’s h, but no v 2: E2 = Ug2 + K2 Since still some h, but also v 3: E3 = K3 Since no more h*, and all v 4: E4 = Ug4 Since no v, and back at original h – but is it? * if RL is defined as top of metal track at bottom ramp L of C of E says:E1 = E2 = E3 = E4 So in the ideal world: Ug1 = Ug2 + K2 = K3 = Ug4 so mgh1 = mgh4 then h1 = h4 But in the real world: Ug1 = Ug2 + K2 + Q2 = K3 + Q3= Ug4 + Q4 so mgh1 = mgh4 + Ffl cosϕ then h1 > h4

23. Swinging Pendulum on a string 2 forms of ME that total amt of NRG fluctuates between max K at equilibrium (vertical) position max Ug 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 total amt of NRG fluctuates between max K at the equilibrium (midpoint) position max Us at the bottom and some at the top max Ug at the top if real world: loss of ME to internal friction inside spring & air resistance

24. Mutli-Color Tracks Which shape track wins? Blue one, with its dip 1st, wins – gets to a faster speed earlier in its trip regardless of real vs ideal… Which shape track gets the ball going the fastest by the end? Ideal: All have = Ug at start, so = K at end, so = v too Real: The red shortest distance (straight) track loses least K to IE, so it’s going a little faster & the blue track deals with most air resistance, since its got high speed for longer, so it’s going a bit slower

25. The Work– Energy Theorem The connection between Work & Energy: work is the act of changing an object’s NRG, while the presence of energy means work can get done. Eq’n: W = ΔE But recall, work is usually done to change an object’s position… so then: W = ΔUg = mgΔh or W = ΔUs= ½k(xf2- xi2) change an object’s motion… so then: W = ΔK = ½m(vf2- vi2) OR possibly more than one, in a complex situation.

26. 2 Categories of Force Conservative Forces – the amount of work done by them does not depend on the length of the path taken Gravity Elastic Electric If only these, then it’s “Physicsland” & ME is conserved Non Conservative Forces - work done does depend on path (aka dissipative forces – cause the amount of ME to change) Friction (air resistance) Tension in a cord Applied forces by a person, animal These lead to thermal energy “losses” – real world.

27. The Math of Conservation of Energy Draw a diagram of “i” & “f” situation use “fast marks” (____) to indicate speed mark RL to indicate height use spring coils (____) to indicate elastic nrg ID any other givens & unknown Decide if any non conservative forces are acting. If no: 2. state: Emechi = Emechf (for ideal - no WNC) 3. then: Uig + Uis+ Ki = Ufg+ Ufs + Kf so are any of the terms = 0? 4. then: mghi + ½kxi2 +½mvi2= mghf+ ½kxf2 + ½mvf2 where mass can cancel if no Us watch units (kg, m, s  Joules)

28. The Math of Conservation of Energy Ex: P#32 Jane, looking for Tarzan, is running at top speed (5 m/s) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does Jane’s mass affect the answer? Does the length of the vine affect the answer? hf= 1.27 m, no, no

29. The Math of Conservation of Energy OR, if dealing with the real world Decide if any non conservative forces are acting. If yes: 2. state: Ei = Ef 3. then: Emechi+ WNC= Emechf(for real - with WNC) Consider a roller coaster as an example: WNC could be friction along the track so Wf = Ffltrackcos 180° so a loss and WNC could be a chain pulling it along track so Wchain = Fchainlchaincos 0° so a gain

30. The Math of Conservation of Energy Ex: P#51 A 47 kg skier traveling 11 m/s reaches the foot of a steady upward 19° incline and glides 15 m along this slope before coming to rest. Determine the amount of friction she runs into along the slope. Ff = 39.6 N

31. 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 = ΔK FΔx cosΦ = ½m(vf2 - vi2) So as v increases, both K and the other side of the equation so goes up by the square of that increase, Ex: for 2x’s as much v, there’s 4x’s as much K 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!