1. Physics 218 Lecture 11
Dr. David Toback
Physics 218, Lecture XI

2. Chapters 6 & 7 Seven Cont Work Work and Energy Last time: This time
Work using the Work-Energy relationship
Potential Energy
Conservation of Mechanical Energy
Physics 218, Lecture XI

3. Physics 218, Lecture XI

4. Physics 218, Lecture XI

5. Kinetic Energy and Work-Energy
Energy is another big concept in physics
If I do work, I’ve expended energy
It takes energy to do work (I get tired)
If net work is done on a stationary box it speeds up. It now has energy
We say this box has “kinetic” energy! Think of it as Mechanical Energy or the Energy of Motion
Kinetic Energy = ½mV2
Physics 218, Lecture XI

6. Work-Energy Relationship
If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes)
Equivalent statement: If there is a change in kinetic energy then there has been net work on an object
Can use the change in energy to calculate the work
Physics 218, Lecture XI

7. Summary of equations
Kinetic Energy = ½mV2
W= DKE
Can use change in speed to calculate the work, or the work to calculate the speed
Physics 218, Lecture XI

8. Multiple ways to calculate the work done
Multiple ways to calculate the velocity
Physics 218, Lecture XI

9. Multiple ways to calculate work
If the force and direction is constant
F.d
If the force isn’t constant, or the angles change
Integrate
If we don’t know much about the forces
Use the change in kinetic energy
Physics 218, Lecture XI

10. Multiple ways to calculate velocity
If we know the forces:
If the force is constant
F=ma →V=V0+at, or V2 - V02 = 2ad
If the force isn’t constant
Integrate the work, and look at the change in kinetic energy
W= DKE = KEf-KEi
= ½mVf2 -½mVi2
Physics 218, Lecture XI

11. I can do work on an object and it doesn’t change the kinetic energy.
Quick Problem
I can do work on an object and it doesn’t change the kinetic energy.
How? Example?
Physics 218, Lecture XI

12. How do you solve Work and Energy problems?
Problem Solving
How do you solve Work and Energy problems?
BEFORE and AFTER
Diagrams
Physics 218, Lecture XI

13. Before and After diagrams
Problem Solving
Before and After diagrams
What’s going on before work is done
What’s going on after work is done
Look at the energy before and the energy after
Physics 218, Lecture XI

14. Before…
Physics 218, Lecture XI

15. After…
Physics 218, Lecture XI

16. Compressing a Spring A horizontal spring has spring constant k
How much work must you do to compress it from its uncompressed length (x=0) to a distance x= -D with no acceleration?
You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?
Physics 218, Lecture XI

17. Potential Energy Things with potential: COULD do work
“This woman has great potential as an engineer!”
Here we kinda mean the same thing
E.g. Gravitation potential energy:
If you lift up a brick it has the potential to do damage
Physics 218, Lecture XI

18. Example: Gravity & Potential Energy
You lift up a brick (at rest) from the ground and then hold it at a height Z.
How much work has been done on the brick?
How much work did you do?
If you let it go, how much work will be done by gravity by the time it hits the ground?
We say it has potential energy: U=mgZ
Gravitational potential energy
Physics 218, Lecture XI

19. Mechanical Energy
We define the total mechanical energy in a system to be the kinetic energy plus the potential energy
Define E≡K+U
Physics 218, Lecture XI

20. Conservation of Mechanical Energy
For some types of problems, Mechanical Energy is conserved (more on this next week)
E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick
K2+U2 = K1+U1
Conservation of Mechanical Energy
E2=E1
Physics 218, Lecture XI

21. E = K + U = ½mv2 + mgy Problem Solving
What are the types of examples we’ll encounter?
Gravity
Things falling
Springs
Converting their potential energy into kinetic energy and back again
E = K + U = ½mv2 + mgy
Physics 218, Lecture XI

22. BEFORE and AFTER diagrams
Problem Solving
For Conservation of Energy problems:
BEFORE and AFTER
diagrams
Physics 218, Lecture XI

23. We drop a ball from a height D above the ground
Quick Problem
We drop a ball from a height D above the ground
Using Conservation of Energy, what is the speed just before it hits the ground?
Physics 218, Lecture XI

24. Next Week Reading for Next Time:
Finish Chapters 6 and 7 if you haven’t already
Non-conservative forces & Energy
Homework 5 Due Monday
Start working on HW6
Physics 218, Lecture XI

25. End of Lecture Notes
Physics 218, Lecture XI

26. Compressing a Spring A horizontal spring has spring constant k
How much work must you do to compress it from its uncompressed length (x=0) to a distance x= -D with no acceleration?
You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?
What is the speed if there is friction with coefficient m?
Physics 218, Lecture XI

27. Roller Coaster A Roller Coaster of mass M=1000kg starts at point A.
We set Y(A)=0. What is the potential energy at height A, U(A)?
What about at B and C?
What is the change in potential energy as we go from B to C?
If we set Y(C)=0, then what is the potential energy at A, B and C? Change from B to C
Physics 218, Lecture XI

28. Kinetic Energy Kinetic Energy = ½ mV2
Take a body at rest, with mass m, accelerate for a while (say with constant force over a distance d). Do W=Fd=mad:
V2- V02 = 2ad= V2
ad = ½V2
W = F.d = (ma) .d= mad
mad = ½ mV2
W = mad = ½ mV2
Kinetic Energy = ½ mV2
Physics 218, Lecture XI

29. Work and Kinetic Energy
If V0 not equal to 0 then
V2 - V02 = 2ad
W=F.d = mad = ½m (V2 - V02)
= ½mV2- ½mV02 = D(Kinetic Energy)
W= DKE
Net Work on an object (All forces)
Physics 218, Lecture XI

30. A football is thrown
A 145g football starts at rest and is thrown with a speed of 25m/s.
What is the final kinetic energy?
How much work was done to reach this velocity?
We don’t know the forces exerted by the arm as a function of time, but this allows us to sum them all up to calculate the work
Physics 218, Lecture XI

31. Example: Gravity
Work by Gravity
Physics 218, Lecture XI

32. Potential Energy in General
Is the potential energy always equal to the work done on the object?
No, non-conservative forces
Other cases?
What about for conservative forces?
Physics 218, Lecture XI

33. Who hits the bottom with a faster speed?
Water Slide
Who hits the bottom with a faster speed?
Physics 218, Lecture XI

34. Mechanical Energy Consider a Conservative System
Wnet = DK (work done ON an object)
DUTotal = -Wnet Combine
DK = Wnet = -DUTotal
=> DK + DU = 0 Conservation of Energy
Physics 218, Lecture XI

35. Conservation of Energy
Define E=K+U
DK + DU = 0 => (K2-K1) +(U2-U1)=0
K2+U2 = K1+U1
Conservation of Mechanical Energy
E2=E1
Physics 218, Lecture XI

36. Conservative vs. Non-Conservative Forces
Nature likes to “conserve” certain types of things
Keep them the same
Kinda like conservative politicians
Conservationists
Physics 218, Lecture XI

37. Conservative Forces
Physics has the same meaning. Except nature ENFORCES the conservation. It’s not optional, or to be fought for.
“A force is conservative if the work done by a force on an object moving from one point to another point depends only on the initial and final positions and is independent of the particular path taken”
(We’ll see why we use this definition later)
Physics 218, Lecture XI

38. Closed Loops Another definition:
A force is conservative if the net work done by the force on an object moving around any closed path is zero
This definition and the previous one give the same answer. Why?
Physics 218, Lecture XI

39. Is Friction a Conservative Force?
Physics 218, Lecture XI

40. Integral Examples we know…
Physics 218, Lecture XI

41. Work done to stretch a Spring
Physics 218, Lecture XI

42. Robot Arm A robot arm has a funny Force equation in 1-dimension
where F0 and X0 are constants.
What is the work done to move a block from position X1 to position X2?
Physics 218, Lecture XI

43. Stretch a Spring
A person pulls on a spring and stretches it a from the equilibrium point for a total distance D. At this distance the force required to keep the spring stretched is F.
How much work is done on the spring in terms of the given variables?
Can we use F.d?
Physics 218, Lecture XI