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CME 100/ENGR 154A

CME 100/ENGR 154A

Ordinary Differential Equations for Engineers

Fall 2016

Prof. Hung Le Problem Set 1

Due: Friday, October 7

1. (5+5 points) The velocity of the current in a river is ~c = 0.6~i+0.8~j km/hr. A boat moves relative

to the water with velocity ~v = 8~i km/hr.

(a) What is the speed of the boat relative to the riverbed?

(b) What angle does the velocity of the boat relative to the riverbed make with the vector ~v?

What does this angle tell us in practical terms?

2. (10 points) A 100-meter dash is run on a track in the direction of the vector ~v = 2~i + 6~j. The

~ is 5~i + ~j km/hr. The rules say that a legal wind speed measured in the direction

wind velocity w

of the dash must not exceed 5 km/hr. Will the race results be disqualified due to an illegal wind?

Justify your answer.

3. (5+5+5+5+5 points) Given the points P = (1, 2, 3), Q = (3, 5, 7), and R = (2, 5, 3), find:

(a) A unit vector perpendicular to a plane containing P, Q, R.

(b) The angle between PQ and PR.

(c) The area of the triangle PQR.

(d) The distance from R to the line through P and Q.

(e) The distance between the plane formed by PQR and point (1, 2, 1)

4. (10 points)The operator?s manual for your lawnmower says ?tighten the spark plug to 15 ft-lb.?

If you are installing the plug with a 10.5-in socket wrench that places the center of your hand 9

in from the center of the spark plug, how hard should you pull at 90° with regards to the handle?

(use units of pounds in your answer)

???

5. (15 points) Find the volume of the parallelepiped formed by the following vectors where AB

???

and AC form the base. Then verify your results using Matlab commands.

???

AB = (?1, 3, 0)

???

AC = (?1, 0, 2)

???

AD = (1, 0.5, 6)

You must submit the Matlab command script for full credit. 6. (25+5 points) The cycloid is the curve traced out by a point on the circumference of a circle

of radius a when the circle rolls along a straight line in its own plane as shown below. This

curve has a number of interesting properties. Among them is the fact that a bead released from

rest in a gravitational field travels from point A to point B in minimum time if its trajectory is a

cycloid.

(a) In this problem your job is to determine a set of parametric equations for x and y as a function of ?. The easiest way to approach this is to realize that the trajectory of point P consists

of the horizontal motion of the center of the circle C and the motion of point P around the

???

???

???

center. Construct two vectors, OC and CP , then add the two to get OP = (x, y). Show

that the parametric equations for the cycloid are given by the following relations:

x = a(? ? sin ?)

y = a(1 ? cos ?) P ? C O

(b) Using Matlab, make a plot of the cycloid over the range 0 ? ? ? 4? with a = 1. Label

your axes and include a title. You must submit the plot and Matlab command script for

full credit.

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