Work, Energy, and Power

1. Work, Energy, and Power § 6.2–6.4

2. Kinetic Energy Energy of a moving mass § 6.2

3. CPS Question To accelerate an object from 10 to 20 m/s requires more work than to accelerate from 0 to 10 m/s. the same amount of work as to accelerate from 0 to 10 m/s. less work than to accelerate from 0 to 10 m/s.

4. Work of Acceleration To accelerate to speed v with constant force F v = F area = Dd slope = a = speed v m t t time • Work = F·Dd F = m (slope) = mv mv 1 1 Dd = vt t t 1 2 2 mv2 • Work = vt = 2

5. Another Perspective So, for the 0–10 vs. 10–20 m/s case: If same force, thensame time a’s and Dv’s are equal, so Dt’s are equal Average speeds are 5 vs. 15 m/s The 10–20 m/s case travels 3 as far

6. A Moving Object Can Do Work Source: Griffith, The Physics of Everyday Phenomena

7. Kinetic Energy the work to bring a motionless object to speed mv2 1 2 K = equivalent to the work a moving object does in stopping

8. Which has more kinetic energy? A. B. C. D. CPS Question 10 m/s 5 kg 10 m/s 5 kg 40 m/s 10 kg 10 kg 20 m/s

9. Rebounding Ball What is the sign of the work done on the ball by the wall as it slows (squishes)? Positive (W > 0). Negative (W < 0). Zero (W = 0). Can’t tell (W = ?).

10. Rebounding Ball What is the sign of the work done on the ball by the wall as it rebounds (expands)? Positive (W > 0). Negative (W < 0). Zero (W = 0). Can’t tell (W = ?).

11. Happy/Sad Balls Which ball has the greatest change in kinetic energy DK during impact? The happy (rebounding) ball. The sad (dead) ball. Both had the same DK.

12. Happy/Sad Balls Which ball has the most (largest absolute value) work done on it during impact? The happy (rebounding) ball. The sad (dead) ball. Both had the same W.

13. Worksheet Problem 1 A luge and its rider, total mass 90 kg, emerges onto a level track with v0 = 36 m/s. It undergoes a constant deceleration of 2.0 m/s2 until it stops. • What is the magnitudeof the force acting on it? • What distance does it travel while decelerating? • First find the general kinematic formula for distance Dx traveled in stopping from speed v0 at acceleration a. • Then find Dx in this case. • What work does the force do?

14. Worksheet Problem 1 A luge and its rider, total mass 90 kg, emerges onto a level track with v0 = 36 m/s. It undergoes a constant deceleration of 4.0 m/s2 until it stops. • What is the magnitudeof the force acting on it? • What distance does it travel while decelerating? • Use the general formula found earlier. • What work does the force do? • What was the initial kinetic energy K0 of the luge?

15. CPS Question The piglet has a choice of three frictionless slides to descend. Along which slide would the piglet finish with the highest speed? A B C The final speed is the same for all.

16. Worksheet Problem 2 A piglet slides down a frictionless ramp of height H and angle a above the horizontal. What is its speed vf at the end? • Find the general kinematic equation for vf when accelerating from rest at acceleration a through a distance Dx. • Geometrically find a and Dx in terms of H and a. • Find vf in this case. • What is the dependency of vf on a?

17. CPS Question The piglet has a choice of three frictionless slides to descend. Along which slide would the piglet finish soonest? A B C The time is the same for all.

18. CPS Question Now the piglet/slide interface has a little friction. Along which slide would the piglet finish with the highest speed? A B C The final speed is the same for all.

19. Work-Energy Theorem • If an amount of work W is done on an otherwise isolated system, the system’s energy changes by an amount DE = W. • The net work done on an object equals its change in kinetic energy Wtot = DK.

20. Work in General curving paths, changing forces § 6.3

21. What’s the point? • What is work when force is not constant or the path is not straight?

22. F may vary with position or time. • In general, dW = F·ds. • So, W = F·ds. Work in General • For constant force, W = F·s. • Path may not be straight. • (Sum of work done over each interval.)

23. Elastic Force Stretching and squishing § 6.3

24. compression tension Structure of Solids • Atoms and molecules connected by chemical bonds • Considerable force needed to deform

25. small stretch larger stretch Elasticity of Solids Small deformations are proportional to force Hooke’s Law: ut tensio, sic vis (as the pull, so the stretch) Robert Hooke, 1635–1703

26. Hooke’s Law Formula F = force exerted by the spring k = spring constant; units: N/m; k > 0 x = displacement from equilibrium position negative sign: force opposes distortion F = –kx

27. backward forward What direction of force is needed to hold the object (against the spring) at its plotted displacement? forward backward • Forward (right). • Backward (left). • No force (zero). • Can’t tell. CPS Question Spring’s Force Displacement

28. slope = k kx area = w force x displacement F = kx • Work = F·x ; • Work = = 1 1 kx·x 1 2 2 kx2 2 Work to Deform a Spring • Push or pull a distance x from equilibrium

29. CPS Question A spring with force constant k is stretched from x = 0 to x = D. What is the work done by the spring as it stretches? 1/2 kD2. –1/2 kD2. 0. It cannot be determined. None of these.

30. CPS Question Two springs, one with a spring constant k1 and the other with a spring constant k2 = 2 k1, are slowly stretched to the same final tension. Which spring has more work done on it? The stiffer spring (k = 2 k1) The softer spring (k = k1) The same work was done on both.

31. Worksheet Problem 3 A 1.53-kg block is released from rest just atop a relaxed spring with k =2.50 N/cm. The block compresses the spring 12.0 cm before momentarily stopping. • While the spring is compressing, what work is done on the block: • by gravity? • by the spring? • What force does the spring finally exert? • What is the block’s finalacceleration? • What is the block’s initialacceleration?

32. Centripetal Force work of acceleration § 6.3

33. Group CPS Question A toy of mass m moving at constant speed v in a circle of radius r has a constant magnitude of centripetal acceleration of v2/r. Its velocity reverses every half-cycle. How much work does the centripetal force do on the toy every half-cycle? mv2. –mv2. pmv2. None of these.

34. 1/2 mv2 = 1/2 mvv is a scalar Kinetic Energy and Direction • K Depends on speed • Direction of velocity is irrelevant • Changing direction only requires force, but no work.

35. Net Force and Net Work • Net force is nonzero if a body accelerates • net work is nonzero if a body changes speed • The net force must overlap with the displacement to do work!

36. Worksheet Problem 4 Your cousin Throckmorton, m = 20 kg, plays on a R = 1.5-m swing. What is the net work done on him as he swings from an angle q = p/6 from vertical down to q = 0? • What is the net force as a function of q? • How far (total path length) does he travel? • Set up the integral for the total work done on him. • Evaluate the total work.

37. Power how quickly work is done § 6.4

38. w = Dt DE Power = Dt Power Rate of doing work DE=change in energy( = work) Dt=time interval

39. DE P = Dt Energy kg m2 kg m2 W = = time s2 s s3 Units of Power = J/s = W = watt

40. W F·Ds P = = = F·v Dt Dt F=force Ds=displacement v=velocity Power A different but equivalent formula

41. CPS Question Dragging a box across a level floor against friction at 1 m/s requires a power of 20 W. How much power is required to drag the same box at 2 m/s? 10 W. 20 W. 40 W. 80 W.