Solutions

1. Solutions

2. 1.a. b. c. 2.

3. 3.

4. vertical forces balance 4. static friction becomes the centripetal acceleration

5. 5.

6. forces at top of loop are normal force and weight, both pointing downwards 6a. sum of forces makes centripetal force as v decreases, n decreases also n = 0 corresponds to minimum v

7. 6b.

8. 6c.

9. 7. & center to center distance altitude from surface, subtract the radius of the Earth from R

10. 8.

11. If an apple has a weight of 1.00 N, then it means that the Earth exerts a 1.00 N force on the apple through gravity. By Newton’s 3rd law, the apple also exerts an equal but oppositely directed force back on the Earth. The apple exerts a 1.00 N force on the Earth. 9. See your notes for details on how the torque depends on force, distance from pivot, and direction angle. 10.

12. 11.

13. 12.

14. 12. continued keep + solution only, x must be positive from the 2.00 kg mass

15. 13.

16. 14.

17. 15.

18. 15. continued Find net torque about the left end: torques balance vertical forces balance as well

19. 16. torques balance about the pivot point

20. 17. time to fall 2.00 m? how far?

21. 18. {tough problem!} eq. #1 eq. #2

22. 18. continued: use energy conservation to relate the speeds top and bottom: combine with the centripetal force equation:

23. 19. {tough problem!} person leaves ice when n = 0

24. 19. continued: when the normal force is set to zero, get the following from the centripetal force equation substitute this into the energy equation

25. 20.

26. 20. continued: Put the pivot point at the left end and balance clockwise and counterclockwise torques. The tension force makes a counterclockwise torque, the masses make clockwise torques. torques balance vertical forces balance as well

27. 20. continued: The x – components also balance. 21 & 22: See notes.

28. 23. Kepler’s 3rd Law: constant

29. 23. Kepler’s 3rd Law: constant

30. 24. Kepler’s 3rd Law: constant The new period is 27 times its original period.