Advanced Classical Mechanics PHYS 331 Fall 2006
Homework Assignment #5
(due: Wednesday, October 3, 11:00 pm)
1. Molecule: Taylor problem 7.8 (HW#4 probl. 5.) (7P)
2. Newton’s Second Law Lab: (HW#4 probl. 6.) (7P)
m
mw
c
One of the PHYS 211 Labs is about Newton’s second law. The experimental set up
consists of a cart with mass mcon a track, a string which is connected to the cart, and
via a pulley connected to a hanging weight mw(see Fig. above). Assume friction is
negligible.
2a. Determine the acceleration ausing Lagrangian Dynamics.
2b. Determine the acceleration ausing Newton’s second law.
3. Friction: Taylor problem (7.12) (4P)
This problem addresses the case of having a nonconservative force (friction) in addition
to a conservative force. Notice that you derive an equation in cartesian coordinates which
does not give you Hamilton’s principle and thus we unfortunately lost the free choice of
general coordinates.
4. Lagrange’s Equations: Taylor problem (7.13) (10P)
Hint: For your proof you will need that for |~1| 1 and |~2| 1 you can approximate
U(~r1+~1, ~r2+~2) = U(~r1,~r2) + 1·~
∇1U+2·~
∇2U.
5. Inclined Plane: Taylor problem (7.16) (6P)
Keep the moment of inertia as general I, i.e. do not replace Iwith the specific Iof a
uniform cylinder.
6. Pendulum with Pivot Point on Wheel: Taylor problem (7.29) (10P)
Hint: Follow Taylor’s hint: Use a cartesian coordinate system with origin in point O.
Express xand yof the mass as functions of ω,φ,t,Rand l.
7. Bead on Wire: Taylor problem (7.35) (10P)
Hint: The hint of Taylor means simply that you have to be very careful in first finding
x(t) and y(t) expressed with ω , R and φ(and t). Make a good sketch and be careful.
You should obtain from the Euler-Lagrange equation that ¨
φ=−ω2sin φ. Also use that
sin(α) sin(β) + cos(α) cos(β) = cos(α−β).
8. Coin on Cone: Taylor problem (7.38) (10P)
Hints: Notice that you use now spherical coordinates (not cylindrical coordinates as in
class). For finding r0in part b) use the same logic as on page 261 for θ(t). For part c)
put (r(t) = r0+(t)) into your equation for ¨r(t) from part b). This gives you an equation
for (t). Approximate this equation for small (t) (you will use a Taylor series for 1
(1+z)3).