Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Work, power and potential energy Conservation of mechanical energy

2. Reminder … • Exam 2 in class on Thursday • Seating will be posted. Closed book (online formula summary permitted) • Material: forces, Newton’s laws, work, power, kinetic and potential energy • Practice tests – tutorial Wednesday • MPHW4 due Friday 12 noon

3. Work, kinetic energy, and the work-kinetic energy theorem

4. Definition of power Power = Rate at which work is done Instantaneous power =limDt0DW/Dt

5. A sports car accelerates from zero to 30 mph in 1.5 s. How long does it take to accelerate from zero to 60 mph, assuming the power (=DW / Dt) of the engine to be constant? (Neglect losses due to friction and air drag.) 1. 2.25 s 2. 3.0 s 3. 4.5 s 4. 6.0 s

6. Power in terms of force and velocity

7. A locomotive accelerates a train from rest to a final speed of 40 mph by delivering constant power. If we assume that there are no losses due to air drag or friction, the acceleration of the train (while it is speeding up) is 1. decreasing 2. constant 3. increasing

8. Work done by gravity • See that this takes a very simple form • Leads to notion of (gravitational) potential energy • conservation of energy

9. Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. The speed of block A at the bottom of its ramp is 1. less than the speed of block B. 2. equal to the speed of block B. 3. greater than the speed of block B. 4. “Can’t tell.”

10. Work done by gravity N Work W=-mg j. Ds j.Ds is just vertical height drop! Therefore: W=-mgh N does no work ! h Ds mg j i

11. A block is released from rest on a frictionless incline. The block travels to the bottom of the left incline and then moves up the right incline which is steeper than the left side. The maximum height that the block reaches on the right incline is 1. less than 2. equal to 3. greater than the height from which it was released on the left.

12. Solution • Same ! • What is change in kinetic energy on way down? Equals work done by gravity • What is kinetic energy at top ? • What is work done on way up ? From previous result – depends on height only….

13. demo • Ball on double inclined path

14. Curved ramp What is now work done ?

15. Work done on curved path At P small amount of work done DW=-mg j.Ds P Total W=-mg j.Spath Ds W=-mgh! Ds h q

16. Work done by gravity Work done by gravitational force in moving some object along any path is independent of the path depending only on the change in vertical height

17. Hot wheels demo • Final speed of cars does not depend on shape of track – only net change in vertical height.

18. Defining gravitational potential energy Ug is called gravitational potential energy

19. Gravitational Potential Energy For an object of mass m near the Earth’s surface: Ug=mgh h is height above arbitrary reference line Measured in Joules J (like K.E)

20. Total energy for object moving under gravity • W-KE theorem now reads: • D (K+Ug)=0  E=Ug+K=constant • E is called the (mechanical) energy • It is conserved ½ mv2+mgh = constant

21. Pendulum demo • Energy (K+U) should be constant • If pendulum released with zero speed will return to same point (height) with zero speed (ignoring air drag, friction etc)

22. Conservative forces • Potential energies can only be defined for conservative forces • Those forces which do work on a object in moving it from one position to another in such a way that the work done is path independent. • eg. gravity, electric forces, spring forces • Not friction, air resistance

23. Another example: springs … Force F=-kx (Hooke’s law) (x extension)  W=-1/2kx2 Therefore, can define elastic (spring) potential energy U= 1/2kx2

24. Work done by spring Area=1/2 base x height F W=-1/2kx2 F=-kx x

25. (Horizontal) Spring x frictionless table spring 1/2kx2+1/2mv2=constant

26. Motion ? • Imagine release from rest at x=a – what happens ? 1/2ka2=1/2kx2+1/2mv2 or 1/2mv2= 1/2ka2-1/2kx2 • what range does x lie in ? • when is v greatest ? • describe motion …

27. Many forces • For a particle which is subject to several (conservative) forces F1, F2 … E=1/2mv2+U1+U2+ … is constant • Principle called Conservation of total mechanical energy

28. A compressed spring fires a ping pong ball vertically upward. If the spring is compressed by 1 cm initially the ball reaches a height of 2 m above the spring. What height would the ball reach if the spring were compressed by just 0.5 cm ? • (neglect air resistance) • 2 m • 1 m • 0.5 m • we do not have sufficient information to calculate the new height

29. Many particles • When system consists of many particles it is only the sum of all the particles energies which remains constant

30. Summary • Total (mechanical) energy of an isolated system is constant in time. • Must be no non-conservative forces • Must sum over all conservative forces • Must sum over all particles making up system

31. Graphical interpretation

32. Spring again oscillates! motion confined to region below dotted line U(x)=1/2kx2 -a a x

33. Force as slope of potential energy graph U vs x Definition : DU=-FDx or F= - DU/Dx Force proportional to slope of U(x) curve Equilibrium corresponds to F=0 ie zero slope

34. Potential Energy graphs unstable equilibrium U(x) E stable equilibrium true equilibrium x