Physics 105, Spring 2015, Reinsch
Homework Assignment 4 (This is a three page document)
Due Wednesday, February 25, 5:00 pm
Problem 1
You have discovered a moon (mass M ) that is perfectly spherical (radius R), not rotating, and has no
atmosphere. Standing on the moon, you throw a stone horizontally with initial speed vo , releasing it
a height h above the surface, and it lands a distance L from your feet, as measured on the surface of
the moon. Both h and L are much less than R.
If one were to ignore the curvaturepof the surface of the moon and assume a uniform gravitational field
g then one would predict L = vo 2h/g, where g = GM/R2 . The field is actually not uniform, and
the trajectory is actually a section of an ellipse, not a parabola. How large is the error in the formula
for L presented above? Compute this using results for Kepler orbits, rather than integrating a ∆g
along a parabolic trajectory. You can present either the exact result or an approximation; either one
will earn full credit.
Note that we have not mentioned the mass of the stone. As discussed in the text, it is useful to
understand that a certain limit exists, the limit in which the mass of the stone becomes infinitesimally
small compared to the mass of the moon. The trajectory of the stone goes to a well defined limit in
this case. For any nonzero value of the mass of the stone, the stone actually accelerates the moon
and we can easily treat the problem exactly by going to the CM frame using the methods described
in the chapter. However this is cumbersome and it is better to understand this limiting process and
the existence of the limit. In this sense, you can use µ for the mass of the stone in your calculations.
The answer does not depend on µ.
Problem 2
Two particles of mass m1 and m2 move in an otherwise empty space and interact via Newtonian
gravity. The initial positions are r10 and r20 , and the initial velocities are v10 and v20 .
(a) The parameters are such that the masses are gravitationally bound. How would you express this
statement mathematically in terms of the four vectors given above?
(b) How long does it take until the velocity vectors are again equal to their initial values?
Problem 3
A billiard table is made in the shape of an ellipse. The z direction is orthogonal to the plane of the
table. Call the foci F1 and F2 . Prove that when a ball bounces elastically off of the edge the product
of the z component of its angular momentum about F1 and the z component of its angular momentum
about F2 is unchanged. Here we are treating the billiard balls as point masses.
Problem 4
In this problem we study time dependence in Example 8.3. For convenience, we will set δ to zero in
Eq. (8.43). We assume φ as a function of time is given by φ(t) = arctan(vo t/ro ).
(a) Verify that Eq. (8.23) is satisfied.
(b) Verify that Eq. (8.24) is satisfied with U = 0.
Problem 5
Figure 8.5 on page 303 shows a plot of Uef f as a function of r. We will express our results in terms
of γ, ℓ and µ. As explained in the text discussing this figure, we are continuing the discussion of
Example 8.1, and the details of the potential are specified there.
(a) Write out the formula for Uef f .
(b) Find req , the location of the minimum.
(c) There is a horizontal reference line labeled “E < 0” in the figure. We know the exact value of the
period. As discussed in lecture, Eq. (8.54) is exact, and Eq. (8.55) is an approximation. We define
∆E = E − Uef f (req ). Write the exact value of the period as a function of ∆E, γ, ℓ and µ.
(d) Thinking of ∆E as small, approximate your answer to part (c) as a constant plus another constant
times ∆E. Evaluate the constants.
(e) We studied a different 1D potential in Problem 1 of Homework 1, and we saw that the first
correction to the period was a negative constant times ∆E to the first power (there were no fractional
powers of ∆E). How does this statement compare to your answer to part (d)?
(f) Looking at Figure 8.5 you can see that as ∆E gets larger the period becomes very large. A related
problem is that of a bead sliding frictionlessly on a wire bent in the shape of the potential energy
curve and set up vertically in an environment with a uniform gravitational acceleration g.
Let F (ξ) be a function that takes a pure number as its input and returns a pure number. For example,
F (ξ) could be cosh(ξ) or ξ 2 + 7ξ 4 . We require that F (ξ) have a single global minimum. Let h be a
parameter with dimensions of distance and Uo be a parameter with dimensions of energy. Find the
EOM for a particle moving along the x axis in the potential Uo F (x/h). Then find the EOM for a
particle sliding on a wire as described above. The shape of the wire is y(x) = h F (x/h). Note that
the equations of motion are fundamentally different. Can you find an F (ξ) so that one of the two
problems has a period that is independent of amplitude, but the other does not?
Hints for Problem 1
• It is easier to compute the exact result than the approximation. Again, either one will earn full
credit. The approximate result is more illuminating. The hints below are for the approximation.
• In finding the approximate result, you will be studying a certain arccos(ξ). It is better to study
the behavior of the quantity ξ before bringing in a series for the arccos.
2
vo
h
.
and gR
• There are two small parameters in the problem. They are R
2
vo
h
• As you work out a series for ξ, you will see a term proportional to R
times gR
. This one
will result in the simple formula for L given in the problem statement.
v2 2
h
h 2 vo2
o
term
and
an
term.
• The next order in smallness has an R
gR
R
gR
• The series expansion for arccos(ξ) will be tricky because you will be expanding about a very
unfortunate point. This can be done by bringing
√ in fractional powers. Alternatively, you can
use the fact that if cos β = 1 − δ, then sin β = 2δ − δ2 . This is useful when δ is small.
• As a motivational remark, you may wish to ponder the following question. If you throw one
stone with h = 10 m and vo = 20 m/s and another stone with h = 40 m and vo = 10 m/s, they
would land at the same spot according to the simple formula for L. Do they really land at the
same spot?