Angle of Flight

Date: 11/27/97 at 20:58:59
From: Joel
Subject: Physics

A fly is sitting at 12 o'clock on a frictionless clock. He takes a 
step to one side and begins to slide down the clock. At what angle, 
relative to 12 o'clock, does the fly "fly" off the clock? (assume he 
doesn't use his wings).

My math teacher gave us this one for fun, even though it's really a 
physics problem. I think the angle I need to find is where the normal 
force is equal to the centripetal force, but I can't get to an 
equation where enough of the variables cancel each other out.  Any 
tips on how to go about this question would be great.  I really want 
to find out the answer and my math teacher says he doesn't remember 
enough physics to solve it.

Thank you :)

Date: 12/01/97 at 09:22:19
From: Doctor Mark
Subject: Re: Physics

Hi Joel,

You're right--this is really a physics problem. In answering this, I 
will have to assume that you know some physics.

Suppose the fly has traversed an angle A from the vertical. That is, 
A is the angle (taken from the center of the clock) between the 
position of the fly on the circumference of the clock, and the 
vertical. We'll also let the radius of the circle (i.e., the clock) 
be R (though it will turn out not to matter what R is).

If you set up a force diagram for the fly, you find that there are two
forces on it: the normal force N (which points radially outward from 
the center of the clock), and the gravitational force, mg, (where m is 
the mass of the fly) which points straight down. Now as long as the 
fly is still traveling along the circumference of the clock, the fly 
is undergoing circular motion, so the net force on the fly must be 
(mv^2)/R, directed toward the center of the clock.

Now if you set up a coordinate system at the fly's position so that 
the x axis points along a tangent, and the y axis points radially 
outward, then we see that (taking components)

   mg cosA - N = (mv^2)/R,   where N is the normal force.

Now the condition for the fly to "fly" off is that it not be in 
contact with the surface. This means that N must be zero! Thus, we 
find the condition to be

       mg cosA = (mv^2)/R.

Now we need to find v.  To do this, we need to use conservation of 
energy. This will give

   total energy at the top = total energy at the position of the fly

where total energy = gravitational potential energy + kinetic energy 
of the fly.

If we take the gravitational potential energy (PE) to be 0 at the 12
o'clock position, then the gravitational potential energy at the 
position of the fly will be

   - mg*(vertical distance the fly has moved).

From the geometry of the situation (you really need to draw a picture
here!), we see that the vertical distance the fly has moved is

   R - R cosA = R(1 - cosA)

so the gravitational PE of the fly will be:

   - mgR(1 - cosA)

The total energy of the fly at the top is

   [PE at the top (= 0)] + [KE at the top 
   (= 0, since the fly starts from rest)] = 0+0 = 0.

The total energy at the present position of the fly is

[PE at the present position of the fly] + [KE at the present position 
of the fly]

   = [- mgR(1-cosA)] + [(mv^2)/2]

Equating the two total energies, we get:

   0 = (mv^2)/2 - mgR(1 - cosA) =====>  (mv^2)/2 = mgR(1 - cosA).

Now comes the mathematics! We want to solve the following pair of
equations for cosA:

   mg cosA = (mv^2)/R

  (mv^2)/2 = mgR(1 - cosA)

To do so, it is most elegant to solve the second equation for mv^2, 
then substitute into the first equation:

      mv^2 = 2mgR(1 - cosA),

whence we find:

   mg cosA = [2mgR(1 - cosA)]/R = 2mg(1 - cosA).

Divide both sides by mg to find that

      cosA = 2(1 - cosA),

whereupon one finds that cosA = 2/3.  Then use a calculator to find 
the angle A.

Now that you know the angle A, you have to convert that to the proper 
time on the clock.  That's pretty easy, isn't it?  I find 22 seconds 
after 1:36 P.M.

This problem is often given as involving a person sliding down an 
igloo. The most interesting thing about the answer is that it doesn't 
depend on the mass of the object doing the sliding, and it also 
doesn't depend on what g is (as long as g is not 0--why?). That means 
that the fly will "fly" off the clock at the same angle whether the 
clock is on the earth, the moon, or on Uranus (which would give an 
interesting twist to the question: What time is it?  "Let me look.")

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 12/01/97 at 18:18:37
From: Wiskey123
Subject: Re: Physics

Thank you for your help :)