E155B - Stanford University



Vector Calculus for Engineers

CME100, Fall 2004

Problem Set #1

(Vector Algebra in 2D and 3D, plotting using MATLAB)

Date: 9/29/2004 Due: 10/6/2004

Reading: Thomas 9.1-9.2, 10.1-10.2

Exercises:

Section 9.1: p. 726 Exercises 14, 16, 34, 40, 54

Section 9.2: p. 735 Exercises 2, 10, 16, 32

Sections 10.1-10.2: p. 805 Exercises 4, 15, 30, 44

MATLAB Workbook (optional):

Exercises 1-6

Problem 1 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 θ. 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, [pic]and [pic], then add the two to get [pic]. Show that the parametric equations for the cycloid are given by the following relations:

[pic]

[pic]

b) using MATLAB, make a plot of the cycloid over the range [pic]with [pic]. Label your axes, include a title (think of your own title), and a set of gridlines.

Problem 2 An electrical circuit is powered by a solar cell as shown below. The cell generates voltage at its terminals that varies as a function of current. The manufacturer of the cell has performed measurements and recorded the following values of the output currents and the corresponding voltages:

|I (Amps) |V (Volts) |

|0.0 |12.0 |

|0.5 |11.5 |

|1.0 |11.0 |

|1.5 |10.5 |

|2.0 |10.0 |

|2.5 |9.5 |

|2.7 |8.5 |

|2.8 |7.0 |

|2.9 |4.0 |

|3.0 |0.0 |

You are an engineer whose task is to estimate the resultant voltages and currents when the cell is connected to a number of different loads: [pic]2, 5, 10, and 15 [pic] by writing Kirchoff’s voltage law [pic]and solving it graphically for each of the load values. [Hint: plot [pic]using the data in the table, then plot [pic]on the same set of axes. Determine the values of the voltages and currents at the points where the “load” lines intersect the “operating curve” of the cell]

-----------------------

[pic]

y

x

O

P

qð

C

θ

C

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download