1. Homework Statement
Bacteria use swimming to seek out food. Imagine that the bacterium is in a region of low food concentration. For the bacterium to profit from swimming to a region with more food, it has to reach there before diffusion of food molecules makes the concentrations in the...
1. Homework Statement
Consider a phospholipid bilayer membrane consisting of a mixture of 90% uncharged lipid and 10% singly charged acid lipid. Assume 68 Å^2 surface area per lipid head group, and assume further that the charged lipids are uniformly distributed and immobile. The membrance...
1. Homework Statement
Derive
z = <L>/Ltot = (2sinhßfa)/(4+2coshßfa)
In the small-force limit the force-extension curve is linear, that is, in this regime the polymer behaves like an ideal Hookean spring with a stiffness constant k [is proportional to] kT/(Ltot*a). Demonstrate this claim...
1. Homework Statement
Consider a protein sphere with a radius of 18Å, and charge Q = -10e, in an aqueous solution of c = 0.05M NaCl at 25 ˚C. We consider the small ions as point charges and use the linear approximation to the Poisson-Boltzmann equation.
What is the surface potential of the...
Besides that when I solve for both A' and B' in terms of C and try to evaluate the ratio, I find
(k-k1)/(k+k1)
which is fine for talking about ratios, but when asked for a probability I want to give a numerical answer.
Not the right section of the forum? Sorry; where should I have posted it?
The ratios are the ratios of reflected to initial, and transmitted to initial. This sounds like, then, the probability of the wave to be reflected and the probability of the wave to be transmitted. But if so, why am...
1. Homework Statement
Consider particles incident (in one dimension) on a potential energy step with E>U.
(That is, particles of total energy E are directed along in one dimension from a region of U=0 to a region of E>U>0.)
Apply the boundary conditions for \Psi and d\Psi/dx to find the...
1. Homework Statement
Consider particles incident on a potential energy step with E<U.
(That is, a particle with total energy E travels along one dimension where U=0, then crosses, at point x=0 into a region where U>E.) (The particle is incident on the potential energy step from the negative...
1. Homework Statement
Consider a lamina rotating freely (no torques) about a point O of the lamina. Use Euler's equations to show that the component of \omega in the plane of the lamina has constant magnitude.
[Hint: Use the reults of Problems 10.23 and 10.30. According to Problem 10.30...
1. Homework Statement
In an electrom microscope we wish to study paprticles of diameter about 0.10 µm (about 1000 times the size of a single atom).
What should be the de Broglie wavelength of the electrons?
Through what voltage should the electrons be accelerated to have that de Broglie...
1. Homework Statement
A triangular prism of mass M, whose two ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis on the z axis. Find its moment of inertia for rotation about the z axis. Without doing any integrals write down and...
1. Homework Statement
Suppose rocket traveler Amelia has a clock made on Earth. She flies to and back from a planet 12 light-years away (as measured from rest with respect to Earth) from Earth at a speed of 0.6c. Every year she sends a signal to Earth. How many signals does Earth receive by...
1. Homework Statement
Gamma-rays can reach a radiation detector by Compton scattering from the surroundings. This effect is known as "backscattering." Show that, when E>>mc^2, the backscattered photon has an energy of approximately 250 keV, independent of the energy of the original photon...
1. Homework Statement
Find the de Broglie wavelength of an 8 MeV proton.
2. Homework Equations
y = h/p
where y is the wavelength
p = mv
K = Eo/sqrt(1-v^2/c^2)
where Eo is the rest mass and K is the kinetic energy
3. The Attempt at a Solution
I make the assumption that "an 8...
1. Homework Statement
h/(mc) = ?
where m is the mass of an electron
2. Homework Equations
3. The Attempt at a Solution
h = 4.136*10^-15 eV*s
m = 0.511 MeV/c^2
c = 299792458 m/s
So m*c (as long as I remember to multiply by 10^6) gives units of eV/c
So h/(mc) gives units of...
1. Homework Statement
A particle moves around the surface of an upside-down cone, in a horizontal circular path, in equilibrium. The particle is given a small radial kick. Use the Lagrangian equation for motion (found in a previous section of this problem):
m\ddot{r} =...
1. Homework Statement
Consider a system of one generalized coordinate theta, having the following Lagrangian equation of motion:
r and b are constants
m is mass
(1/3)mb^{2}\ddot{\theta} = r(r+b)\theta + r^{2}\theta^{3} + gr\theta
And this potential energy (if it matters):
U =...
This is what I'm trying to do:
(b) Find the two equations of motion. [Got it.] Interpret the ø equation [\dot{\phi}mr^2 = 0] in terms of the angular momentum l_{z}, and use it to eliminate \dot{\phi} from the r equation in favor of the constant l_{z}.
First off, what is l_{z}?
That's the...
Oh, well, my equations of motion were, for r and phi, respectively,
m(r-doubledot) = rm(phi-doubledot)^2
and
(d/dt)mr(phi-dot)=0
So the r-component of momentum is m(r-dot),
and the phi-component of momentum is (phi-dot)mr
?
The form of the r-component is familiar to me, thinking of p=mv and...
That's great news! I had always seen spherical coordinates written with the theta and phi's meaning opposite of this.
Now, then, I need to find out how to get the z-component of the angular momentum. How can I do this?
L = r X p --- in Sphereical Coordinates
1. Homework Statement
Express in vector form angular momentum in spherical coordinates.
\rho is the distance from the origin to the point
\phi is the angle made in the Cesarean x-y plane counter-clockwise from the positive x-axis
\theta is the...
1. Homework Statement
A particle is confined to move on the surface of a circular cone with its axis on the vertical z axis, vertex at the origin (pointing down), and half-angle a.
(a) Write the Lagrangian L in terms of the spherical polar coordinates r and ø.
(b) Find the two equations...
1. Homework Statement
A hard rubber cylinder of radius r is held fixed with its axis horizontal, and a wooden cube of mass m and side 2b is balanced on top of the cylinder, with its center vertically above the cylinder's axis and four of its sides parallel to the axis.
Assuming that b < r...
1. Homework Statement
Given a potential energy as a function of r, make a phase diagram for r>0.
2. Homework Equations
3. The Attempt at a Solution
I didn't write down the exact function for this, since this isn't a problem I have to do, but the thing is is I don't know how to...