PHYSICS 321 - Classical Mechanics I

Spring 2007

Homework Set #8 (due 3/28/07)

1. Thornton and Marion, problem 5-1.

2. Thornton and Marion, problem 5-2. Hint: Use the differential form of Gauss’ Law, which is Eqt. 5.37 in the text book. To calculate the divergence of a vector field in spherical coordinates, see Eqt. F.18 in Appendix F. Check your answer using the Shell Theorem.

3. Thornton and Marion, problem 5-3.

4. (a) Thornton and Marion, problem 5-7. To do the integral, look at Appendix E.

(b) From your result in part (a), use the Taylor series expansion to evaluate the potential very close to the rod, i.e. in the limit that R<<l. (You should expand the quantity (1 + (2R/l)²)^1/2 to first order in (2R/l)².) Then take the gradient of the potential to find the gravitational field. The result should look familiar from electrostatics – the electric field of an infinitely long charged rod.

5. Thornton and Marion, problem 5-15. Hint: First use the Shell Theorem to calculate the gravitational force on the mass as a function of its distance from the center of the Earth. Then use Newton’s 2^nd Law to obtain the equation of motion of the mass.

6. Thornton and Marion, problem 5-16. Use the method of Example 5.4 in the book. This integral you can do yourself. Check your answer by using Gauss’ Law to calculate the gravitational field g due to the infinite sheet of mass. From that you can find the force on the sphere. By Newton’s 3^rd Law, the force on the sphere is equal and opposite to the force on the sheet.