Merced College; Don Power

Lab 3 -- Helix NAME _________________________

Your assignment is to write a vector equation ( and the equivalent set of parametric equations) that models the helix on a cardboard tube (such as a paper towel tube, wrapping paper tube, toilet paper tube, etc.).

Your equation must be correct to scale, with the parameter t representing the total angle in radians, all distances (for the x-, y- and z-coordinates of points on the curve) given in millimeters, and a range of values for t that equates to the full length of the tube.

Your curve should have:

a. Its axis along the positive z-axis

b. Starting point (when t=0) on the positive x-axis (for CCW curves) or y-axis (for CW curves)

c. Make t = 2p at one complete revolution (when the curve is directly above the starting point).

Part 1. Measurements:

Give me the following measurements (to the nearest millimeter) so that I can properly evaluate your equation(s):

a. Radius of tube _________________

b. Length of tube _____________________

c. Length (measured parallel to the axis) of one circuit along the helix. _________________

d. Stand the tube on a flat surface. Does the curve turn clockwise or counterclockwise as it rises?

(Clockwise/counterclockwise) ____________________

Part 2. Calculation Results:

Your equation in vector form (exact values, not rounded): r = ______________________________

Your set of equations in parametric form:

x = ______________________

y = ______________________

z = ______________________

Also calculate (give rounded values, correct to 4 significant digits):

1. Angle of intersection between the curve and the base (in degrees): ______________________

2. Arc length of the curve: __________________________

3. Curvature at the point where t = 2p: ________________________

4. Equation (solved for z) of the osculating plane at the point where t = 2p:

____________________________________________________

Part 3. Computer Graph.

Graph your curve using the tubeplot command in Maple. Before turning in the lab results, you should verify that your parametric equations for the helix actually result in a graph with the correct orientation and dimensions.

Note: I want you to do the calculations for this entire lab using vectors. However, you should be aware that almost the entire lab can be done with right triangle trigonometry, if you visualize what you get when you unroll the tube. This would be a good way to check some of your basic calculations.

Return to:  Merced College; Don Power               Updated 07/10/06 by Don Power