1. Topics for Today Lab workbooks are available at the bookstore
Finish up conservative forces (8-2)
Elastic potential energy (8-1)
Reading a potential energy curve (8-3)

2. Conservative Forces
Quiz – Two identical balls roll down two different tracks with the same starting and ending heights. The first track is straight. The second track curves up and down. How does the speed of the ball at the end of the first track compare with the speed of the ball at the end of the second track?
faster
same
slower
Ignore energy losses due to friction or drag.

3. Conservative Forces
Quiz is the force conservative?
Yes No

4. Elastic Potential Energy
Recall that: ∆𝑈=− 𝑥 𝑖 𝑥 𝑓 𝐹 𝑑𝑥
For a spring, we define x so x = 0 and U = 0 at the relaxed position of the spring. The force is F = -kx. Inserting these into the equation above:
𝑈 𝑥 =− 0 𝑥 −𝑘𝑥 𝑑𝑥 =𝑘 0 𝑥 𝑥 𝑑𝑥 = 𝑘 𝑥 2
𝑈(𝑥)= 𝑘 𝑥 ← Elastic potential energy
Spring stores potential energy whether stretched or compressed.

5. Spring Force
Demo 1M Spring Gun and Darts use 7 gm and 28 gm darts
Quiz – Putting a dart into the dart gun compresses a spring in the gun by a certain distance. When the spring is released it does work on the dart. Now use a second dart with one quarter the mass. How high will the second dart go?
1. Same as the first ball
2. Twice as high as the first ball
3. Half as high as the first ball
4. Four times as high as the first ball

6. Potential Energy Curves
Recall that: ∆𝑈=− 𝑥 𝑖 𝑥 𝑓 𝐹 𝑑𝑥
Let’s take the derivative of both sides:
𝑑 𝑑𝑥 𝑈=− 𝑑 𝑑𝑥 𝐹𝑑𝑥 → 𝐹 𝑥 =− 𝑑𝑈(𝑥) 𝑑𝑥
Check using elastic potential energy:
𝑈 𝑥 = 𝑘 𝑥 2 → 𝐹 𝑥 =− 𝑑𝑈(𝑥) 𝑑𝑥 = − 𝑘 𝑑 𝑑𝑥 𝑥 2 = −𝑘𝑥

7. Potential Energy Curves
For spring:
𝑈 𝑥 = 𝑘 𝑥 𝐹 𝑥 =− 𝑑𝑈(𝑥) 𝑑𝑥 = −𝑘𝑥

8. Potential Energy Curves

9. Turning Points Mechanical energy: 𝐸=𝑈 𝑥 +𝐾(𝑥)
Solve for K: 𝐾 𝑥 =𝐸− 𝑈 𝑥 = 𝑚 𝑣 2
What happens at points where K(x) = 0?
Have v = 0, so particle cannot move past.
These are “turning points” – the particle slows down to v = 0 and then turns around.
By plotting the total mechanical energy on a potential energy curve, we can immediately find the turning points.

10. Turning Points
If a particle starts at large x, moving left with E = 5 J, how far can it go?
Answer 1-5 for x1 to x5.
Where will the particle be going the fastest?

11. Equilibrium Points Recall: 𝐹 𝑥 =− 𝑑𝑈(𝑥) 𝑑𝑥
Places where dU/dx = 0 have force = 0.
These places are called “equilibrium points”.
There are three kinds of equilibria:
Stable – if the object moves slightly to either side, the forces push the object back to the equilibrium point.
Unstable – forces on either side push the object away from the equilibrium point.
Neutral – there are no forces on either side.

12. Equilibrium Points Quiz – responses are:
1=Stable, 2=Unstable, 3=Neutral
What sort of equilibrium will there be at x4?
At x3?