[SOLVED] Physics 342 Homework 8

1: Easy
A) Calculate the magnitude of the centrifugal acceleration due to the Earths rotation at the
equator, at a latitude of 45 degrees and at the north pole. Notice, this represents a local correction
to gravitational acceleration that varies with latitude. Comment on the direction of this fictitious
force. I want your answer in m/s2. Notice, this is a correction to the local direction of ~g.B) There is also a centrifugal acceleration due to our orbit around the sun, calculate its magnitude and compare it to the centrifugal force due to the Earths rotation.C) Calculate the speed you would need to be going in a car when turning in a circle of radius
100 meters that would produce an acceleration on the driver equivalent to the acceleration due to
gravity near the surface of the Earth. Report this speed in terms of miles per hour.D) What velocity would you need to be traveling at for the magnitude of the coriolis force to
equal the magnitude of the centripetal force at a latitude of 45 degrees. Decompose your velocity as
using the local plane geometry of figure 10-9. Pick which direction you should move to maximize
the coriolis eect and put all of your velocity in that direction.Write your answer in miles per hour.Hint to calculate the MAGNITUDE at 45 degrees you do not need to form a vector cross
product. Remember what the cross product means and look at figure 10-3
2: (short) You find yourself in a spinotron the amusement park ride where you stand against a
wall and the whole thing begins to rotate. Once it is rotating fast enough the floor drops out from
under you and you find yourself suspended in midair.There are 4 forces acting on you friction,
the normal force of the wall, gravity, and the centripetal force. Free body the equilibrium situation
and determine the time it takes for one complete rotation of the ride such that you are held in
place against the wall neither slipping rising or falling o.I want a purely algebraic answer. What
happens to you if the ride slows down or speeds up its down rate from this equilibrium spin rate?Start with the equation at the end of example 1.8 and set P F~external = mA~ Youll get an equation
in z and r for force and acceleration, solve them and then tell me what happens if ! = increases
or decreases.
3: (long) Imagine a ball is set on a merry go round which is rotating counterclockwise at a
fixed angular frequency !. The ball starts at x0 and y0. Determine the equation of motion if only
coriolis acceleration comes into play. Remember, the rotation is in the z direction and the motion
is only in the x and y direction.See example 2.10 for help. Allow for arbitrary x0 and y0. Set
V~ (0) = vx0
i + vy0 j. Pick values for your angular frequency and initial velocities and initial positions
to plot ~r(t). Youll need to pick numbers for initial conditions and solve for all the constants in ~r(t).This and the next problem are essentially two dimensional x and y problems. Neglect gravity, the
normal force, and friction. Do this in the rotating frame, see equation 10.30.
4: (Long) Repeat Problem 3:, this time do it for both the centripetal force and the coriolis force.
Assume. Youll find your equations are coupled in x and y. Try to combine the two equations into a
single equation using the complex variable = x + iy.When you have combined the two equations
into one equation in terms of , , , etc. This will be another heavily weighted problem. Graph
your solution for various choices of the spin rate ! and various initial conditions. Do this in the
rotating frame, see equation 10.30. Give me the full solutions for x(t) and y(t).Hint add the the ax to i ay and determine a formula of the form a + b + c = 0. Attempt
to find solutions proportional to Aet and determine by direct substitution. You should find a
repeated root in which case (t) = (A + Bt)eit By inspection A = x0 + iy0 and B = vx0 + ivy0.Equate the components and describe the subsequent evolution, graph it in x and y if you can.
5: A projectile is launched directly upwards with an initial vy = vy0. Determine an expression for
its east/west deflection due to the coriolis force. Your answer should be in terms of , !, g, and vy0.This has essentially been done in the book and in class. Plug in real numbers for as high as you
could throw a ball and a missile launched vertically to a height of 10,000 meters launched at Grand
Junctions latitude.
6: A missile is fired due north at a latitude of 40 degrees south. and longitude 120 degrees
east. The launch angle is 35 degrees with respect to the horizon. The initial velocity of the shell
is 1000 m/s. What is the total deflection of the missile due to the coriolis force? Hint calculate
the coriolis deflection velocity and the time before the projectile lands.The total deflection is then
just vcoriolis tland. Please note that the solution to a very similar problem in the book is wrong. If
you use the solution Ill know and this will be given a grade of zero. ! may be decomposed as
!(cos()i + sin()j).Your flight time should be 117 seconds, get this to get the deflection.
hint use this geometry
7: (Longish but graded as such) Model a toilet bowl as a circular truncated cone with r1 = 1 f t
and r2 = r1/4 When you flush the toilet water begins to fall at approximately the free fall rate
(gravitational rate vz = gt).A) Now, the toilet will drain when all the water goes through the exit which can only happen
at a rate of M = watervwaterAreadrain (take this as given from fluid dynamics).The total time for a
toilet to drain is then given by RMdt = mwaterinitiallyintoilet. Make some reasonable assumptions about
the mass of water in a toilet and estimate the time it takes for a toilet to drain. Go flush a toilet
and see if your number is remotely reasonable. The velocity here can be approximated as |v| = |gt|.Why is your answer too small? What happens that we neglect to account for? Go flush a question
and answer this. Ive given you water draining from a tank, not what happens in a toilet. Make an
analogy to what you observe and what you learned in the chapter on why planets orbit.B) Treating a single particle of H2O as a point mass falling through a toilet height h in a time
t given by your estimate above calculate the eastward/westward deflection and velocity in said
direction due to the coriolis eect. Also calculate the magnitude of the maximum coriolis force per
unit mass on a parcel of water falling at the velocity it reaches when it is going down the drain.Specifically, give me the z magnitude of F~coriolis/m.
Hint If youre clever there is no new calculus here, look at example 10.3. Does it make a big
dierence if you use twice the time you calculated in part 1? 5 times the time?C) Compare this force to the normal force acting on a molecule of H2O sitting on a flat surface.
Compare the two forces. Now, think about this. When a ball rolls down an inclined plane the
normal force of the plane is taking the force due to gravity and re-directing the motion of the ball
so that it rolls down at whatever the plane angle is.If you look at a toilet it is the geometry of the
bowl and the resulting normal force which controls the sense of rotation.There IS a size of a tank
of water for which the coriolis force is appreciable assuming no curvature. What would change if
I replaced the toilet with one of the big pools at Sea World draining through the same size drain?
What Changes?
8: A huge bucket of water is set spinning about its center. Determine the equilibrium profile of
the water, specifically z(h). There is a pressure force, a centrifugal force, and a gravitational force.Free body the gravitational versus centrifugal force to determine what the pressure force MUST
look like. Recall F~p + m~g + F~centripetal = 0 More specifically, determine the angle between mg and
mr!2 and notice that the tangent of this angle is dz
dr. Either that or google equilibrium height of
rotating fluid.What curve represents this shape? This is actually how people spin mirror surface
when making telescopes using parabolic mirrors.