Homework 1
Math 352, Fall 2011
Instructions: Solve at least two of the following three problems. Your solutions must be
written in LATEX.
Due Date: Friday, September 9
1. A bar of length 4π pivots around the unit circle, as shown in PivotAnimation.gif.
(a) Find parametric equations for the spiral traced out by the endpoint of the bar.
(b) Find the length of this spiral between the points (1, 0) and (1, −4π).
2. A square of side length L rolls along the bottom of the catenary y = cosh x, as shown
in RollingSquare.gif.
(a) Find parametric equations for the path taken by the center of the square.
(b) For what value of L is this path a straight line?
3. For each t ∈ [0, 1], let L(t) be the line segment with endpoints (t − 1, 1 − t) and (t, t).
The following picture shows L(t) for several different values of t:
1
0.5
0
-1
-0.5
0
0.5
1
Let R be the region covered by the segments L(t) as t ranges from 0 to 1. Then R has
three sides: two straight sides on the left and right, and one curved side on the top.
What is the shape of the curved side? Justify your answer.