fromthenotes

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From the notes: Two masses on a string going through a hole

Two masses on a string going through a hole.  Yeah, that's the stuff we learn about in physics class.  Anyway, this is the first in a (possibly) series of posts where I work out some problem from my lecture notes or homework.  The reason for doing this is because 1) there's a (small) chance it will be useful or interesting to someone else, and 2) the drawn out process of typing this up and thinking about how to explain it helps me study.

Figure 1, two masses on a string going through a hole.

 

Consider two masses connected by a string (figure 1).  The ideal string passes through a hole in a plate which is parallel to the x-y plane.  One mass, is sitting on the plate and is free to rotate about the hole with no friction.  The second mass, , is hanging below the plate and moves only in the vertical z direction under the influence of gravity.   The total length of the string is and we define and to be the distances between the hole and and , respectively, so that .

We can use Lagrangian mechanics to explore the system.  In Cartesian coordinates, the total kinetic energy is the sum of the kinetic energy of each mass:

     (1)

where we've used the dot notation for time derivatives.

Considering that is only rotating about the axis and is moving only vertically, it makes sense to consider switching to a more reasonable coordinate system.  We can transform to cylindrical coordinates with:

Taking derivatives of these with respect to time and inserting them into (1), we can rewrite the kinetic energy in cylindrical coordinates as

where and are now the generalized coordinates of the system.  The potential energy depends only on the height of since is confined to sit on the plate, so

With and we can write the Lagrangian :

We see that does not appear in the Lagrangian, which means that the generalized momentum corresponding to is a conserved quantity:

     (2)

can be identified as the angular momentum of mass .

We can also look at the total energy :

     (3)

From (2) we see that .  Substituting this in (3) gives

We can separate the constants on the left hand side and use the remaining terms to define an effective potential :

Imagine is rotating about the hole.  This means there will be some centripetal (or centrifugal, depending on your reference system) acceleration which will pull the hanging mass up.  We want to find such that the force is an extremum.  In other words:

Setting the derivative equal to zero gives

And solving for :

which gives the radius of "orbit" of mass about the hole for a given angular momentum such that remains suspended in the air.

 

Pretty Graph of the Day

I ended up with this graph when doing some homework:

534HW2p3

It's the xy phase portrait of the system of differential equations as shown at the top of the figure.  There's a critical point at (0,0) and a limit cycle with radius . What a beauty!

The graph was made with pplane.

Fun with Diffraction Gratings

A laser beam passing through a transmission diffraction grating straight on gives the standard diffraction pattern we all know and love.  It's a bit more interesting, though, if the beam hits the grating at an angle:

grating1

figure 1

Most intro optics books cover this situation, and the result is (equation 1):

where is the grating spacing, is the angle of the mth maxima,  is the incident angle, is the maxima order, and  is the wavelength. We can rewrite this (homework) in a more useful way as (equation 2):

A similar, but slightly more complicated situation happens when you rotate the grating instead of the lazer:grating

figure 2

With a rotated grating (figure 2), the laser is still hitting the grating at an angle as it is in figure 1.  So, starting with equation 2 and using some geometry we get the angles that satisfy the maxima condition:

This seemed liked a fun thing to model in Mathematica, especially since I had never played with any of the graphics features before.



You can view the source here.

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