1. Instructor: Dr. Tatiana Erukhimova
Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 22, 23, 24

2. Mechanical energy is conserved!
Work-energy theorem:
Mechanical energy is conserved!

3. Examples Strategy: write down the total mechanical energy, E,
E = KE + U
at the initial and final positions of a particle:

4. Initial E1=KE1+U1…

5. Final E2=KE2+U2

6. Then use or

7. H

8. Who hits the bottom with a faster speed?
Water Slide
Who hits the bottom with a faster speed?

9. Roller Coaster
You are in a roller coaster car of mass M that starts at the top, height H, with an initial speed V0=0. Assume no friction.
What is the speed at the bottom?
How high will it go again?
Would it go as high if there were friction? H

10. Roller Coaster with Friction
A roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2).
Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

11. Conservative Forces
If there are only conservative forces in the problem, then there is conservation of mechanical energy
Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another
Good examples: Gravity and Springs
Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost.
Good example: Friction (like on Roller Coasters)

12. Law of Conservation of Energy
Mechanical Energy NOT always conserved
If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc.
Energy = Kinetic Energy + Potential Energy + Heat + Others…
Total Energy is what is conserved! K1+U1 = K2+U2+EHeat…

13. Total Energy is what is conserved! K1+U1= K2+U2+EHeat…

14. Force of gravity:
Potential energy function:
Spring:
Potential energy function:

15. Spring problem revisited
A block of mass M is on a horizontal surface and is attached to a spring, spring constant k. If the spring is compressed an amount A and the block released from rest, how far from unstretched position will it go before stopping if there is no friction between the block and the surface?
How will this answer change is the block is not attached to the spring??

17. A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude Vm. What is magnitude of the velocity when the bullet returns to the ground? How high will it go?

18. Quiz
A block of mass m is placed against a vertical spring, spring constant k. The spring is unstretched at y=0. A
If the spring is compressed an amount A and the block released from rest, how high will it go?

19. Quiz
A block of mass m is attached to a vertical spring, spring constant k. The spring is unstretched at y=0. A
If the spring is compressed an amount A and the block released from rest, how high will it go?

20. Block of mass m has a spring connected to the bottom
Block of mass m has a spring connected to the bottom. You release it from a given height H and want to know how close the block will get to the floor. The spring has spring constant k and natural length L. H
y=0

22. ENERGY DIAGRAMS

23. Potential Energy Diagrams
For Conservative forces can draw energy diagrams
Equilibrium points
Motion will move “around” the equilibrium
If placed there with no energy, will just stay (no force)

24. Stable vs. Unstable Equilibrium Points
The force is zero at both maxima and minima but…
If I put a ball with no velocity there would it stay?
What if it had a little bit of velocity?

25. A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in the Figure. The particle is released from rest at point A.
What is the direction of the force on the particle when it is at point A?
At point B?

26. c) At what value of x is the kinetic energy of the particle a maximum?
d) What is the force on the particle when it is at point C?
e) What is the largest value of x reached by the particle during its motion?
f) What value or values of x correspond to points of stable equilibrium?
g) Of unstable equilibrium?

27. 2 or 3D cases: If or
then

28. Several dimensions: U(x,y,z)
Partial derivative is taken assuming all other arguments fixed
Compact notation using vector del, or nabla:
Another notation:

29. Geometric meaning of the gradient
Direction of the steepest ascent;
Magnitude : the slope in that direction
Direction of the steepest descent
Magnitude : the slope in that direction

30. Given the potential energy function
find the x and y components of the corresponding force.

32. Have a great day! Reading: Chapter 9
Hw: Chapter 9 problems and exercises