PH 314 Homework Fall 2003

Set 1 Due Monday September 8, 2003
 

• 1. Turn in the spreadsheet for a body falling due to air resistance with F_v = -0.05 v^2, for 5, 10, and 20 sec. (See instructions on course webpage)
• 2. Prob 1.34, p. 35. Most of the credit will be on part c)
• 3. Prob 1.41, p.36

• 1. Do a numerical calculation in Excel of a block sliding smoothly on a sphere. Find the angle the block leaves the surface if it starts from rest at a 10 degree angle from the vertical. Repeat for 20 degres from the vertical. Email in the sheet.
• 2. Prob 1.37, p. 36 [ The second part, finding the maximum range, will count extra.]
• 3. 2.18 p. 69

• 1. Create a spreadsheet for a sphere subject to air resistance and wind. Use a radius of 3.6 cm, mass of 150 g, and a drag coefficient of 0.15 instead of the 1/4 in the text. Add a scroll bar (slider) to control wind velocity. Find the range of a sphere launched at 60 degrees above the horizontal with a speed of 45 m/s with no wind, and with a wind of 8 m/s in the negative x direcion. Email in the sheet.
• Instructions for installing a slider can be found on the PH 314 web page.

2. Create a spreadsheet for motion of a charge in subject to electric and magnetic fields. Let m=1e-8 kg and q = 1e-6 kg. Use Ex = 10 V/m, Bz = 0.05 tesla. Start the mass at the origin with no velocity and integrate over at least one full cycle of the motion. Save the graph and email it in. Now let the initial y-velocity be 100 m/s,and email in the graph of this motion.
• 3. Taylor prob 4.3
• 4. Taylor prob 4.10

• 1. Consider the frequencies of oscillation of a 1-kg mass in the following 1-D potential

U(x) = 3*sin(x) + 2/(x+0.2) + x, in joules.
a) Find the potential minima accurately (start with a plot to get your bearings).
b) Find the frequency of small oscillations around each minimum between x=0 and x=5.
• 2. Taylor, 4.24
• 3. Taylor, 4.30
• 4. Taylor, 4.34

1. A 250 kg mass is to be dropped from 0.4 m onto a (massless) platform sitting on a spring whose force constant is 17000 N/m. You are to
install a dashpot (damping force proportional to velocity) so that the platform will come to rest without overshoot in the shortest possible time.
Determine the theoretical damping constant for the dashpot. (Note: this requires something which will be superior to critical damping.)
2. Model the above situation via RK2 in a spreadsheet. Do one plot of critical damping, and another where you can adjust the critical damping.
Obtain a  numerical value for the damping coefficient by eyeballing the graph.
3. Turn in your analysis of the RLC circuit. Q values from measurements, inductance L, and series resistance R.

1. Turn in the dimensions of your homemade coupled pendulum setup. Sketch the setup. Say what you used for masses. Report the pendulum frequency and how it compares to theory. Report the time for energy exchange between masses and report how it compares to theory.

2. Work out the tabletop problem with a string length of 2.5 m, and a hanging 2.0-kg mass and a 1.0-kg mass on the tabletop. Let the initial angle be zero, the initial rdot = +0.6 m/s, initial r = 1.0 m, and initial thetadot = 0.5 rad/sec. Calculate the max and min r values theoretically, solving a numerical equation for r.

3. Do an RK2 simulation of this problem, and obtain numerical values of rmax and rmin. Email me the spreadsheet.
 

1.Turn in your analysis of the coupled oscillation problem. M1 = 0.2178 kg, M2 = 0.2839 kg,  freq with m2 and spring: 0.52 Hz. Find the oscillation frequencies of the two modes, and determine the amplitude ratios for each mode.
2. Show that the total energy E in an elliptical orbit is -k/(2a), where k is the gravitational coefficient, and a is the semi-major axis of the ellipse. [Hint: look at r_max and r_min.]
3. Then show that the semi-minor axis is given by b = L/sqrt(2 mu |E|). [For this, you may use the result of #2. The numerator of this expression is the angular momentum, L, not 1.]

1,2.  Create a spreadsheet for the inverted driven pendulum. Let the pendulum be a rod of length 0.30 m, driven at one end. Launch it a 0.1 radians and find a; driving frequency and amplitude for it to be stable inverted. Measure the period of the stable motion. Calculate the period from the formula, and compare to your simulation.
3.Create a spreadsheet for a Foucault pendulum of length 20.0 m.  In order to see anything in a few swings, you will have to greatly increase the Earth's rotational rate. Your graph should show rotation of the plane of oscillation. Calculate the rotation rate, and measure it from your simulation.

1. Model a water molecule having a mass 16 u at the origin, a mass u a distance L along the x-axis, and a mass u at a distance L from the
origin at an angle of 110° fromthe x-axis.
a) Calculate the CM of this system.
b) Calculate the inertia tensor for the water molecule in a coordinate system centered on the molecule's CM, with axes parallel to the x and y
axes.
2. Perform a similarity transformation on the inertia tensor by going to a coordinate system rotated by 55°. This should result in a diagonal tensor,
giving the principal moments of the water molecule. (The principal axes are supposed to lie along the symmetry directions of the body.)
3. The fractional difference in the Earth's moments of inertia is about 1/300, the Earth being slightly flattened at the poles. Use this
information to calculate the rate at which the Earth's angular velocity vector precesses (wobbles) about the direction of the Earth's
rotational angular momentum (you could say this is the rotational North pole)
4. Consider a uniform rectangular plate of sides a and b, which is forced to rotate about an axis through a diagonal of the plate with an
angular velocity W.
a) Show that the principal moments of inertia of this plate are (M/12) a^2, (M/12)b^2, and (M/12)(a^2+b^2).
b) Identify the direction of torque needed to rotate the plate. (Give this in relation to the plane of the plate.)
c) Find the magnitude of the torque, in terms of M, a, b, and W.
5. Calculate the landing point for a projectile launched (ignore air resistance) from the N pole 1500 km due South at an angle of 45 degrees. Use the coriolis correction to find its displacement, to first order in the angular velocity of the Earth.. (You can also independently check this.)

Friday November 14, 2003 (optional)

1. Start with the lagrangian function for the two-body problem 8.20, and obtain the Hamiltonian, and the two hamiltonian equations of motion.
2. For the football pictures, write out the relations for the moments about 1 and 3 axes, use theta = 22 degrees, use 'measured' phidot, and psidot, and obtain a numerical estimate of  I_12 / I_3.