EXAM I, PHYSICS 4304 - Physics and Astronomy | Physics and ...



EXAM II, PHYSICS 4304

April 3, 2002

Dr. Charles W. Myles

INSTRUCTIONS: Please read ALL of these before doing anything else!!!

PLEASE write on one side of the paper only!!

PLEASE do not write on the exam sheets, there will not be room! If you do not have paper, I will give you some.

PLEASE show all of your work, writing down at least the essential steps in the solution of a

problem. Partial credit will be liberal, provided that the essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.

NOTE!!! the setup (THE PHYSICS) of a problem will count more heavily in the grading than the detailed mathematics of working it out.

PLEASE clearly mark your final answers and write neatly. If I cannot read your answer, you can't expect me to give it the credit it deserves and you are apt to lose credit.

PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!

NOTE!!!! Work any three (3) of the five problems. Each problem is equally weighted and worth

33 points for a total of 100 points on this exam (you get 1 point for signing your name!).

1. A mass m is confined to the x-axis. It experiences a conservative force which is derivable from a potential energy function given by (k and ( are positive constants):

U(x) = (kx2 – ((x4

a. If the mass has total mechanical energy E, find an expression for the velocity v at position x. (Find v(x). This will not have the time in it!) (5 points)

Find the zeros of U(x). Find the equilibrium points. Determine whether these are points of stable or unstable equilibrium. Use these results to sketch the potential U(x). (6 points)

b. Make a sketch of the phase diagram for this nonlinear oscillator for the following total mechanical energies: E = (k2/(, E = ( k2/(. (6 points)

c. Calculate the restoring force F(x) and (by combining this with Newton’s 2nd Law) derive the differential equation of motion for this mass. (Don’t try to solve it yet!) (4 points)

d. Suppose the term – ((x4 in the potential is much smaller than (kx2, so that the mass is almost a harmonic oscillator. With initial conditions that at time t = 0, the mass is at rest at x = A, use the method of successive approximations (described next) to find an approximate solution, x(t), to the nonlinear equation obtained in d. As the initial approximation, take the harmonic oscillator solution (obtained when ( = 0) for x(t) which satisfies these initial conditions. Use this approximation in the equation of motion to obtain the next approximate solution. Make sure that this solution satisfies the initial conditions! Assume that ( is small enough that this process can be stopped after one iteration. Briefly discuss the resulting solution for x(t) physically. Compare and contrast this non-linear oscillator solution with that for the harmonic oscillator. (12 points)

The following integrals might be useful: (cos(ut)dt = (1/u)sin(ut). (sin(ut)dt = -(1/u)cos(ut).

Don’t forget the integration constant!!!

The following identity might be useful: cos3(x) = (¾) + (¼)cos(3x)

NOTE!!!! Work any three (3) of the five problems.

2. NOTE: Parts a. and b. are (obviously!) independent of each other!

a. By a successive approximation procedure, use your calculator to solve the following equation: x + 1 = 2cos(x).

Obtain a result that is accurate to four significant figures. You may use either direct iteration or Newton’s Method. I strongly recommend Newton’s method! Hint: Your initial approximation should be some x > 0. Before choosing this initial approximation, it might be useful to make a sketch of x+1 and 2cos(x) on the same graph to give you an idea where the two functions cross. (17 points)

b. Using complete and grammatically correct English sentences, define the following terms: deterministic chaos, limit cycle, Poincar( section, mapping, bifurcation, Lyapunov exponents. (16 points)

3. Answer the following questions about an infinitely long right circular cylinder of radius a and uniform volume mass density (. Hint: Recall that, especially for situations of high symmetry, field calculations are often much easier if Gauss’s Law is used! (Hint: I worked an example almost like this in class!)

a. Calculate the gravitational field g outside the cylinder and a distance r ( a from the axis. (8 points)

b. Calculate the gravitational field g inside the cylinder and a distance r ( a from the axis.

(8 points)

c. Calculate the gravitational potential ( at all points in space. Take the zero of potential to be at r = a. (12 points)

d. A point mass m is placed a distance R ( a away from the cylinder axis. Compute the gravitational force between m and the cylinder. (3 points)

e. Compute the gravitational potential energy for the point mass in part d. (2 points)

The following might be useful: Lateral surface area of cylinder of radius r and length (: A = 2(r( Volume of cylinder of radius r and length (: V = (r2(.

Answer the following questions about a sphere of radius R which has a non-uniform volume mass density, which depends on the distance r from the sphere’s center as ( = (0 (r/R)2, where (0 is a constant density. Hint: Recall that, especially for situations of high symmetry, field calculations are often much easier if Gauss’s Law is used!

Calculate the gravitational field g outside the sphere and a distance r ( R from the center. (10 points)

Calculate the gravitational field g inside the sphere and a distance r ( R from the center.

(10 points)

Calculate the gravitational potential at all points in space. Take the zero of potential to be at r = (. (13 points)

The following integral might be useful: (rn dr = (rn+1)/(n+1), where n is any power (n ( -1). Don’t forget the lower limit!!!

NOTE!!!! Work any three (3) of the five problems.

4. NOTE: Parts a., b., c., d. and e. are (obviously!) independent of each other! Work each with Newton’s Universal Law of Gravitation (or results obtained from it in Ch. 5) and not by assuming that the gravitational acceleration g is a constant! Assume that the Earth and the Moon are spheres with uniform mass density. Data needed to obtain the required numerical results in parts a., b., c., and d.: Universal gravitational constant =G =6.67 ( 10–11 N(m2/kg2, Earth Mass = ME = 5.98 ( 1024 kg, Earth Radius = RE = 6.38 ( 106 m, Moon Mass =

MM = 7.35 ( 1022 kg, Moon Radius = RM = 9.3 ( 105 m.

a. A particle mass m is shot from the Earth’s surface at a speed of v0 = 7 ( 103 m/s. Calculate the maximum height which it will reach before beginning to fall back to Earth. A numerical result is wanted & needed! (5 points)

b. A particle of mass m is dropped from rest from a height h = 3 ( 105 m above the Earth’s surface. Calculate the speed with which it will hit the ground. A numerical result is wanted & needed! (5 points)

c. Calculate the escape velocity for a particle of mass m on the surface of the Moon. That is, find the minimum velocity needed to “escape” from the Moon’s gravitational field. A numerical result is wanted & needed! (5 points)

d. A spherical satellite of uniform mass density and mass m = 2000 kg is in a circular orbit of radius r = 4.0 ( 107 m about the Earth. Compute the gravitational force between the satellite and the Earth, the speed of the satellite in orbit, and the period of the satellite’s orbit. Compute the effective gravitational acceleration (in m/s2) experienced by the satellite. Numerical results are wanted & needed! (6 points)

A particle of mass m is dropped from rest from a height h above the Earth’s surface. By using conservation of total mechanical energy, derive an expression for the time t it takes to fall to the Earth’s surface. This will be in the form of a messy integral. Leave it in integral form! (12 points)

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