Ignoring minor issues like friction, if a ball is placed on an uneven surface it
ID: 2840347 • Letter: I
Question
Ignoring minor issues like friction, if a ball is placed on an uneven surface
it will roll in the direction of steepest descent. That is: it will roll in the
direction opposite that which maximizes the directional derivative of f where the graph
of z = f(x,y) describes the shape of the surface.
Suppose we have the surface described by the function:
f(x,y) = (x^2 + y^2) / 20
_______________________________________...
A) Let (x(t), y(t)) be coordinates of the path that the ball would take. Explain why
(x'(t) , y'(t)) = -? f (x(t),y(t))
That is, describe the left and right side in a quantitative sense (what are they measuring?) and see if it agrees.
B) Suppose the ball is at position (x0, y0). Compute ? f and substitute it into the equation above in order to produce two simple differential equations. Solve these two differential equations in order to find the path that the ball will take.
C) Compute the limit as t goes to infinity of your solution. Does this value agree wit your conclusion in the previous exercise?
Explanation / Answer
f(x,y) = (x^2 + 2y^2)/20
You can get an idea of the shape of this surface by plotting the vertical cross-sections in the x-z and y-z planes. For the x-z plane, set y=0 (which is the equation that defines that plane!), and see what you get. For the y-z plane, set x=0, and see what you get. Finally, try several horizontal cross-sections with different values of z (and this is exactly the first thing that (a) asks you to do).
You should see that it's an elliptical paraboloid, opening upward, with its minimum at the origin.
(b) sounds like you'll want to write the equation of motion, using
F_ = ma_ = m d^2 r_/dt^2
where underscore denotes a vector, and
r_ = (x,y,z).
This means finding F_, the resultant of 2 forces --
1. gravity: F_g = -mg e_z, where