SFSU Physics & Astronomy Department
Physics 330 Problem Sets
Problem Set 1:
Problem Set 2:
Hint for Problem 50. The calculation is simpler, and the interpretation of the results clearer, if you change variables, from time t and velocity v(t) to the dimensionless time and speed variables,
Problem 2-17mod. (This is a altered version of problem 2-17 in Marion and Thornton. .
a) A baseball player smacks
the ball directly over home plate, at a position which we shall call x = 0, y =
0. The ball leaves the player's bat at an elevation angle 
above
the horizontal, with an initial speed of v0 = 55 m/s. (This is about
110 mph.) Suppose that the ball lands at a place at the same height as where it
was hit, yfinal = 0.
Show that the distance that the ball travels is
and that the maximum range is
Calculate the numerical value of Rmax.
b) Now let us require that the ball clear a fence, a distance d higher
than the point where it was hit; that is, at yfinal
= d. Show that the distance traveled before just clearing the
fence is given by
.
This is good.
c) Find the numerical values of the maximum range Rmax
and the corresponding launch angle 
,
for a wall 2.0 m higher than the point at which the ball is struck.
Determine the values of Rmax and to
four significant figures.
NOTE: Taking the derivative of R and setting it equal to zero, solving for , and calculating Rmax,
should work. However, it is easier to set up a spreadsheet and calculate R
for a range of values of around 45o. That is
the method that I recommend.
Problem Set 3:
· Problem
2-9. Consider a projectile fired vertically in a constant gravitational
field. For the same initial velocities, compare the times required for the
projectile to reach its maximum height (a) for zero resisting force, (b) for a
resisting force proportional to the instantaneous velocity of the projectile.
The projectile is fired upwards initially.
- Problem
2-13. A particle moves in a medium under the influence of a
retarding force equal to mk(v3+a2v),
where k and a are constants. Show that for any value of the initial speed
the particle will never move a distance greater than p/2ka and that the particle comes to
rest only for t --> infinity.
The way to approach this problem is to set m dv/dt equal to the given force, which is a function of v. Integrating with respect to time is not easy. However, the variable of integration can be converted from t to v by using the trick, shown in the text, of multiplying by v dt on both sides of the equation. Then the integral can be done, leading to the result for the maximum range. For the statement about when the particle comes to rest, you can avoid doing a difficult integral by taking the limiting case of the retarding force for large times, and therefore small velocities. - Problem
2-42. A solid cube of uniform density and sides of length b is
in equilibrium on top of a cylinder of radius R (figure 2-C). The planes
of four sides of the cube are parallel to the axis of the cylinder. The
contact between cube and sphere is perfectly rough. Under what conditions
is the equilibrium stable or not stable?
The cylinder is assumed to be fixed so it can't rotate, so that the block moves away from the equilibrium point on top of the sphere by rolling off to the side. One approach to analyzing conditions of equilibrium is to calculate the potential energy of the block as a function of the angle by which
the cube rotates as it rolls, and to look at the derivatives of the
potential energy with respect to . Another approach is to look for
restoring torques to maintain a stable equilibrium.
Problem Set 4:
- Problem
3-6. Two masses m1 = 100 g and m2 slide
freely in a horizontal frictionless track and are connected by a spring
whose force constant is k = 0.5 N/m. Find the frequency of oscillatory
motion for this system.
The center of mass of this system will remain fixed as the two masses oscillate about it. Each of them will experience the same force as if it was connected by a spring to the center of mass. Figure out the effective spring constant that the mass sees, and you have solved the problem. - Problem
3-7. A body of uniform cross-sectional area A = 1 cm and of
mass density
r
= 0.8 g/cm3 floats in a liquid of density 
= 1 gm/cm3 and
at equilibrium displaces a volume V = 0.8 cm3. Show that the
period of small oscillations about the equilibrium position is given by
,
where g is the gravitational
field strength. Determine the value of 
.
You need to use Archimede's law to find the
buoyant force; it will depend on the vertical position of the "floating
log" in such a way as to lead to the equation for the harmonic oscillator.
Problem 3-8.
Hints on problem 3-8. From the expressions given for x and y you can show that
leading to
And
leads to
Then F = ma gives
m d2s/dt2 = -m(g/l)s
Problem Set 5:
- Problem
3-21.
This problem is easily done with Mathematica. Here is a bit of help.
From x = (A+Bt) exp (-beta t), we get dx/dt = (B-beta A - beta B t) exp (-beta t). We use these as x and y in a parametric plot, with t as the parameter. The example below makes two representative plots. The values of the parameters used are:
|
A |
B |
beta |
x (t=0) |
v (t=0) |
|
1 |
1 |
1 |
1 |
0 |
|
-1 |
-.5 |
1 |
-1 |
0.5 |
- The Mathematica
command is
{beta=1,ParametricPlot[{{(1+1t)Exp[-beta t], (1 - beta 1 - 1 beta t) Exp[-beta t]},
{((-1)-.5 t)Exp[-beta t], (-.5+beta + .5 beta t) Exp[-beta t]}},{t,0,10}]}
This problem can also be done in a spreadsheet. - Problem 3-23. Note that nine plots are required. Erratum: The plot to take as a model should be figure 3-7 (and not figure 3-6). This problem can be done either in a spreadsheet or with Mathematica.
Problem Set 6: 10, 14, 15, 16
- Hints on problem 5-14. The gravitational self-energy is the total energy which would be released if the body was assembled bit by bit by bringing mass in from a large distance away. Calculate the change in potential energy of an element of mass dm brought in from infinity to the surface of a spherical body of radius r, increasing its radius to r + dr. Then integrate.
- Hints on problem 5-15. It is easy to reason that the force on the particle is proportional to the distance from the center of the earth. Then proceed based on the analogy between this situation and that of a harmonic oscillator.
- Hint on problem 5-16. Start with the case of a point mass above a disk discussed in the book and take the appropriate limit. (Applying this to a sphere of finite radius has to be justified too.)
Problem Set 7: Ch.
5 P. 17, 20;
· Hint on problem 6-1. When you set up the integral over the path length, you will have an integrand with a square root, which you can put in the form (1+e)1/2, where as usual e represents a quantity small compared to 1. You can expand this using the binomial theorem: (1+e)1/2 = 1 + 1/2 e - 1/8 e2 + (terms of third order in e). You have to keep the third term, because you will find that the term proportional to a vanishes.
· Hint on problem 6-4. This works out smoothly if you use cylindrical coordinates, (r, q, z), and let the polar angle q be the independent variable.
· Hint on problem 6-10. This problem does not require any use of the calculus of variations. It is just an example of minimizing with a constraint.
Problem Set 8:
Problem Set 9:
- Hints on Problem 7-6. The answer in the back of the book gives the Lagrange equations, with x the horizontal position of the block and S the distance the hoop has rolled down the plane. (I used x instead of x and y instead of S.) "Integrals of the motion" means quantities which are conserved, due to the first term in a Lagrange equation (the partial of L with respect to the generalized coordinate) being zero. In this case, it will be a momentum.
Problem Set 10:
- Hint on Problem 7-18. Please just skip the part about finding the "line about which the angular motion extends equally in either direction . . . ." The answer in the back seems to be correct only in the small-angle approximation, and none of us have found an attractive solution for the more general case.
Problem Set 11:
- Problem 7-27. To make this problem manageable, choose as generalized coordinates
{qi} = xCM, yCM, R, q}
Problem Set 12:
· Hint on Problem 8-1: Note that we cannot now take R = 0.
· Hint on Problem 8-2: Make the change of variable to u = l/r before trying to integrate. Here you may use Mathematica to do the integral. However, please follow these ground rules when using Mathematica to carry out any part of a calculation for you: first put the integral in dimensionless form and make any other obvious changes of variable which simplify the calculation; then either print out the output from Mathematica or copy it verbatim into your solution; and remember that even if you use Mathematica to do part of a problem for you, you should make a full write-up of your solution to the problem, including diagram, discussion of the problem, details of solution, and discussion of the result.
· [Hint on Problem 8-5: Assume a mass m orbiting a much larger mass M to do the calculation; then explain why the "equivalent one-body problem" would lead to the same result even if m and M (or m1 and m2) were similar in magnitude.] (Save for another year.)
· Hint on Problem 8-7: In order to calculate the time averages, first use Lagrange's equations to find the general solution for x(t), y(t), and z(t)).
Problem Set 13. Ch. 8 P. 26, 32, 36, 41