PHYSICS 321 - Classical Mechanics I
Spring 2010
Homework
Set #8 (due 3/31/10)
1. Thornton
and Marion, problem 5-1.
2. Thornton
and Marion, problem 5-2. Hint: Use the
differential form of Gauss’ Law, which is Eq. 5.37 in the text book. To calculate the divergence of a vector
field in spherical coordinates, see Eq. 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 2nd 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 3rd Law, the force on the sphere is equal and
opposite to the force on the sheet.