1. Fall wk 7 – Thus.11.Nov.04 Welcome, roll, questions, announcements Energy, work, and forces Review derivatives Spring workshop Energy Systems, EJZ

2. Energy review Kinetic energy = ½ mv 2 Potential energy due to gravity = mgh (near Earth’s surface) Potential energy due to gravitational attraction between two masses = -GmM/r

3. Conceptest: review kinetic energy

4. Review energy,work, and forces Recall that force = - gradient of potential energy F x = - dU/dx Recall that work = force * displacement. Only the component of force parallel to the displacement does work: W=F. d Conceptests from Calculus Ch.2.4, 2.

5. Conceptest: work and force

6. Work done by constant forces Quick experiment: Pull a mass along the table with force meter Measure force (F) and distance pulled (x) How much work (W) did you do at each point? How does it depend on the mass (m)? Graph F(x) and W(x) Ch.7 #12 (p.160)

7. Work done by variable forces W=F. d assumes that the force F is constant. What if the force varies? Investigate this with springs: 1. Add small masses to spring  calculate U g 2. Measure force F i exerted by each mass increment m i 3. Measure incremental displacements d i of spring 4. Calculate incremental work done W i at each step 5. Graph F i (x), W i (x), U(x) and compare

8. Variable forces: data mimi didi xUiUi FiFi F(x)

9. Variable forces: results F x W x

10. Variable forces: questions 5. Graph F i (x), W i (x), U(x) and compare QUESTIONS: 1.What relationship do you see between the spring force at each point and the work done on the spring by gravity? 2.What is the slope of the F(x) curve? What does it tell you about your spring? 3.How does the potential energy stored in the spring depend on the distance stretched? 4.How could you describe the potential energy stored in the spring, algebraically?

11. Calculating variable forces U spring =1/2 kx 2, F=-dU/dx=____ Ch.7 Fig.7-11 (p.150): Prob.# 27, 28, 61 (p.161) Applying calculus: If U=-GmM/r, find F Ch.8 # 38 (p.192), 72 (p.295)

12. Phys 7 HW 2 Ch.7 #12, 27, 28, 61 Ch.8 # 16, 38, 72 You can now calculate the force due to a known potential energy, using derivatives. After we learn integration in calculus, you will also be able to calculate the potential energy for a known force. This is useful, because it is easier to measure forces, and it is easier to do calculations with energy conservation.