Download Problems - TTU Physics

REVIEW/DIAGNOSTIC HOMEWORK, PHYSICS 5306, Dr. Charles W. Myles
Due, Monday, September 11, 2006 in my office or mailbox by 5pm!
These problems are taken from exams in Physics 4304 (Undergraduate Mechanics) from recent semesters. The
primary purpose of these is to FORCE YOU to review undergraduate mechanics! Yes, there are a lot of problems &
yes some are tedious. However, if you have trouble with these, you should wonder whether or not you should be in
this GRADUATE course! If you have trouble, chances are you'll have difficulties in the rest of the course! If you
have trouble, I strongly suggest that you consider dropping this course & taking Physics 4304 as a "leveling
course". You can take this course again in Fall, 2007. Working these problems is a good time to form Study
Groups to collaborate on homework. Its easier to learn stuff with friends' help! If you don't have friends in class,
make some! In real situations, physicists work in groups!! NO CONSULTATION with people who had this
course previously & NO use of problem solutions posted in previous years is allowed! Problem solutions (.jpg
format) will be posted on the Homework Web Page shortly after the due date. Any cheating will be dealt with
harshly! (HINT: SOME of these problems or variations of them might be found in the undergraduate Mechanics text by Thornton & Marion.)
INSTRUCTIONS: PLEASE write on one side of the paper only!! PLEASE don’t write on the problem sheets,
there won’t be room! 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 essential work is shown. Organized work, in a logical, easy to
follow order will receive more credit than disorganized work. The setup (PHYSICS) of a problem will count more
heavily in than the detailed mathematics of working it out. PLEASE clearly mark your final answers & write neatly.
If I can’t read your answer, you can't expect me to give it the credit it deserves and you are apt to lose credit . Each
problem is equally weighted.
1.
A block of mass m is moving on a horizontal surface where measurements have found that the retarding force is
proportional to the (1/6) (one-sixth) power of the velocity. That is, F = - mkv1/6, where k is a constant. At t = 0,
the velocity is v0 and the mass is at the origin. Find (in any order)
a. The velocity as a function of time (v(t)).
b. The time it takes the mass to stop.
c. The position as a function of velocity (x(v)).
d. The position as a function of time (x(t)).
e. The distance the mass travels before it stops.
The following integral might be useful. The constant of integration is not shown.. The lower limit is important in this problem!
∫vp dv = (vp+1)/(p+1), where p is any power (p  -1)
2.
A block of mass m moves along the x-axis. At time t = 0, it is at x = 0 and its velocity is v0. At t = 0, a time
dependent force given by F = F0e-kt begins to act, where F0 and k are constants. Find (in any order)
a. The velocity as a function of time (v(t)).
b. The position as a function of time (x(t)).
c. Find v(t) and x(t) when t is very small (but NOT zero!) In a few complete and grammatically correct English
sentences, explain what this small time means physically. Do v(t) and x(t) resemble any familiar results in
this limit? Hint: I don’t want you to take the t = 0 limit here! I want you to take the small t limit, keeping
the lowest order time-dependent terms in v(t) and x(t)!
d. Find v(t) and x(t) when t is very large. In a few complete and grammatically correct English sentences,
explain what this long time means physically. Do v(t) and x(t) resemble any familiar results in this limit?
e. When first formulating this problem, I originally was going to ask you to compute the time it takes the mass
to stop. However, I then realized that computing this in the usual way (finding the time t when v(t) = 0) leads to
unphysical results. Despite this, compute this time by this method anyway. In a brief, complete and
grammatically correct English sentence, explain why is this an unphysical result.
The following integral might be useful. The constant of integration is not shown. The lower limit is important in this problem!
∫e-kt dt = (e-kt)/(-k). This expansion might be useful: If |kt| <<1, e-kt  1-kt + (½)(kt)2 –
3.
A mass m is confined to the x-axis by a conservative force which is derivable from a potential energy function
given by (k and ε are positive constants): U(x) = (½)kx2 – (¼)εx4. (This is a non-linear oscillator!). Assume a given
total mechanical energy E.
a. Find a general expression for the velocity v at position x. (That is, find the functional form of v(x). This expression
will also depend on E and will not have the time in it)!
b. For a given E, derive an equation, which when solved will give the turning points of the particle. (Don’t try to
solve this equation for general E!). What is the velocity at these turning points? Solve the equation for the turning
points in the special case when E = 0.
c.
Find zeros & the equilibrium points for this potential. Determine whether the equilibrium points are points
of stable or unstable equilibrium. Also find the values of x where U(x) = 0. Use all of these results to sketch the
potential U(x).
d.
e.
f.
Make a sketch of the phase diagram for this nonlinear oscillator for the following total mechanical energies:
E = (k2)/(8ε), E = (k2)/(2ε).
Calculate the restoring force F(x) & (by combining this with Newton’s 2nd Law) derive the differential equation of
motion for this mass. (Don’t try to solve it yet!)
Suppose the term – (¼)εx4 in the potential is much smaller than (½)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 to find an approximate solution, x(t), to the nonlinear equation obtained in e. As
the initial approximation, take the harmonic oscillator solution (obtained when ε = 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 ε is small enough that
this process can be stopped after one iteration. Briefly discuss (using complete, grammatically correct English
sentences!) the resulting solution for x(t) physically. Compare and contrast this non-linear oscillator solution
with that for the harmonic oscillator. Hint: Part of this discussion should include the fact that there are
terms in x(t) which oscillate at a different frequency than the harmonic oscillator. At what frequency do
these terms oscillate?
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)
4.
A one-dimensional, driven, damped harmonic oscillator has mass m, spring constant k and is subject to a
velocity-dependent damping force given by – bv, where b is a constant and v is the velocity. The oscillator is
underdamped. The driving force is F0 sin(ωt), where F0 is a constant and ω is a frequency.
a. Find the steady state (particular) solution for the displacement x(t). This means derive the results for the
frequency-dependent amplitude D(ω) and phase angle δ(ω). It doesn’t mean copy them directly from a
book!
b. Consider the steady state solution. If β = b/(2m) = (⅓)ω0, where ω0 is the natural frequency, compute the
amplitude D(ω) and the phase angle δ(ω) exactly at resonance. Compute the quality factor Q of this
oscillator.
These identities might be useful: cos(θ  φ) = cosθcosφ -/+ sinθsinφ; sin(θ  φ) = sinθcosφ  cosθsinφ
5.
Consider a one-dimensional periodic force, F(t). Over one time period T, it has the form
F(t)
= - F0, (-½)T  t < 0;
F(t)
= F0, 0 < t  (½)T
where F0 is a constant. Note: T is a time period for which one can define a frequency as ω = (2π/T). Using this fact in your solution
below will be useful!
a.
b.
Find the Fourier series representation of F(t).
Consider a one-dimensional, driven harmonic oscillator with NO DAMPING (b = β = 0). The driving force has the
form F(t), above. Use the known solution for the steady state displacement xs(t) of an undamped oscillator with a
sinusoidal driving force, along with the results of part a, to express the solution for x(t) for this driven oscillator (with
F(t) above) as a series solution.
The following integrals might be useful. The constants of integration are not shown. The lower limits are important in this problem!
∫cos(ax)dx = (1/a)sin(ax)
6.
∫sin(ax)dx = -(1/a)cos(ax)
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 ρ = ρ0 (r/R)2, where ρ0 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!
a.
Calculate the gravitational field g outside the sphere and a distance r  R from the center.
b.
c.
Calculate the gravitational field g inside the sphere and a distance r  R from the center.
Calculate the gravitational potential Φ at all points in space. Take the zero of potential to be at r = .
∫rn dr = (rn+1)/(n+1), where n is any power (n  -1).
Answer the following questions about an infinitely long right circular cylinder of radius a and uniform volume
mass density ρ. Hint: For situations of high symmetry, field calculations are often much easier with Gauss’s
Law!
a. Calculate the gravitational field g outside the cylinder and a distance r  a from the axis.
b. Calculate the gravitational field g inside the cylinder and a distance r  a from the axis.
c. Calculate the gravitational potential Φ at all points in space. Take the zero of potential to be at r = a.
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.
e. Compute the gravitational potential energy for the point mass in part d.
The following integral might be useful (Don’t forget the lower limit!!!):
7.
The following might be useful: Lateral surface area of a cylinder of radius r & length : A = 2πr.Volume of
cylinder of radius r & length : V
8.
= πr2.
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) 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 for 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 (numerically!)
b.
c.
d.
e.
9.
the maximum height which it will reach before beginning to fall back to Earth.
A particle of mass m is dropped from rest from a height h = 3  105 m above the Earth’s surface. Calculate
(numerically!) the speed with which it will hit the ground.
Calculate (numerically!) 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.
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 (numerically!) 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 (numerically!)
effective gravitational acceleration (in m/s2) experienced by the satellite.
A particle of mass m is dropped from rest from a height h above the Earth’s surface. Use
conservation of total mechanical energy to 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!
See Figure. Write the Lagrangian & use Lagrange's Equations to derive the
equations of motion for each of the three systems shown. In each case, also derive
an expression for the generalized momenta, write the Hamiltonian, & derive
Hamilton's Equations of motion. Show that the latter are equivalent to those
obtained by Lagrange's Equations.
a. A mass m is shot vertically upward in a uniform gravitational field.
b. A mass m is tied to a spring of spring constant k and is allowed to vibrate vertically in a uniform
gravitational field.
c. A simple pendulum of mass m and length  is swinging in a vertical plane in a uniform gravitational
field.
10. See Figure. “Atwood’s Machine”: Two masses, m1 & m2 (m2 > m1) are connected by a massless,
inextensible string of length . The string is put over a massless, frictionless pulley so that
both masses hang vertically, as shown. The system is released from rest at time t = 0.
a. Write expressions for the kinetic energy, the potential energy, and the Lagrangian of
the system. How many degrees of freedom are there? What is the constraint? Write
an equation expressing this constraint mathematically.
b. Derive the equations of motion using Lagrange's equations. Find the acceleration of
the system.
c. Use Lagrange's equations with undetermined multipliers to find the generalized
d.
e.
forces acting on the masses. Use this method to determine the string tension S.
Derive expressions for the generalized momenta of the system. Write an expression for the
Hamiltonian for the system.
Derive the equations of motion using Hamilton's equations. Show that this is equivalent to the results
obtained in part b.
11. See Figure. Two identical masses are attached to two identical springs, shown.
The masses move in one dimension on a frictionless horizontal plane.
a. Write expressions for the kinetic energy, the potential energy, and the
Lagrangian of the system. How many degrees of freedom are there?
(Hint: Obviously, the correct potential energy will contain cross terms in the coordinates x1 and x2, so that the equations of motion
for x1 and x2 will be COUPLED!)
b.
c.
d.
Derive the equations of motion using Lagrange's equations.
Derive expressions for the generalized momenta of the system. Write an expression for the
Hamiltonian.
Derive the equations of motion using Hamilton's equations. Show that this is equivalent to the results
obtained in part b.
12. A particle of mass μ moves under the action of an attractive central force, which has a potential energy of the
form: U(r) = -K[(e-ar)/r]. K and a are positive constants. (This is known as the Yukawa Potential & is a crude model of the
potential between protons & neutrons in a nucleus).
a. Derive an expression for the central force F(r).
b. For fixed angular momentum , calculate the effective potential energy V(r), which includes both the
potential U(r) and the so-called “centrifugal” term, which depends on . Sketch of V(r) and qualitatively
DISCUSS the radial motion for various particle energies E. Include cases of both E > 0 and E < 0 in this
discussion!
c. For a particular angular momentum 0, the particle is observed to be in a stable circular orbit of radius r0.
(Recall that a circular orbit means that the total mechanical energy E0 is equal to the minimum value of the effective potential V(r)).
Calculate the angular momentum 0 for this circular orbit.
d. Calculate the total mechanical energy E0 for the circular orbit of part c.
e. Calculate the speed v0 and the period T0 for the circular orbit of part c.
NOTE: The answers to parts c, d, and e should depend on (at most) a, K, μ, and r0.
13. Each of the following problems deals with a particle of mass μ moving in a central force field. Parts a and b are
independent of each other and independent of parts c and d! (HINT: The easiest way to solve these is to use the differential
equation for the orbit, rather than the integral form!)
a.
b.
c.
d.
The particle orbit r(θ) is an ellipse. That is, (α/r) = 1 + ε cos(θ), where α = (2)/(μk) & ε (<1) is the
eccentricity, given by ε2 = 1 + (2E2)/(μk2). Here, E (<0) is the particle energy,  is its angular momentum,
and k is a positive constant. PROVE that the force F(r) has the inverse r-squared form [that is, PROVE that F(r) = -(k/r2)].
The force has the form F(r) = -(K/r3), where K is a positive constant. Calculate the corresponding
potential energy function U(r). For fixed angular momentum ℓ calculate the effective potential energy
V(r). Find the equation for the orbit r(θ).
The particle orbit is given by the equation r = Aθ, where A is a positive constant (geometrically, this is a spiral
orbit). Calculate the force function F(r) and the corresponding potential energy function U(r). For fixed
angular momentum ℓ calculate the effective potential energy V(r).
Calculate r(t) and θ(t) for the particle orbit in part c. In addition to the time t, these will depend on μ, A,
and ℓ. Don’t forget the integration constants!!!
14. This problem requires that you know details about elliptic orbits. In parts a –f, I want NUMBERS with proper
UNITS!! An asteroid, mass m = 6 1020 kg, is in an elliptic orbit about the sun. The orbit eccentricity is ε =
0.75. The semi-major axis of the orbit is a = 5 1011 m. (All distances are measured from the center of the sun.) The sun mass
is
M = 2 1030 kg. Gravitational constant: G = 6.67 10-11 N m2/kg2. Calculate:
a.
The maximum and minimum distances of the asteroid from the sun (aphelion and perihelion), rmax and rmin, and
the semi-minor axis, b, of the elliptical orbit.
b.
c.
d.
e.
f.
g.
The total angular momentum, ℓ, of the asteroid.
The total mechanical energy, E, of the asteroid.
The period of the orbit.
The speed, vmax of the asteroid when it is at perihelion (rmin). Its speed, vmin when it is at aphelion (rmax).
The speed, v of the asteroid when it is a distance r = 3.75 1011 m from the center of the sun.
Write the equation (with numerical factors explicitly evaluated) which describes the orbit r(θ).