21st Century Astronomy: The Solar System (Sixth Edition)
21st Century Astronomy: The Solar System (Sixth Edition)
6th Edition
ISBN: 9780393691283
Author: Laura Kay; Stacy Palen; George Blumenthal
Publisher: W. W. Norton
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Chapter 7, Problem 36QP

(a)

To determine

The density of Vesta.

(b)

To determine

The difference about the composition of Vesta.

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Draw a third quadrant vector C. (remember that boldface characters represent vector quantities). Show the standard angle 0 for this vector (= angle that C makes with the positive x- axis). Also show the angle that C makes with the negative y-axis: call the latter angle 8. Finally, show the smallest angles that C makes with the positive x-axis and the positive y-axis: call these angles p1 and p2, repectively. a) Prove the following formulas for the components of C involving the standard angle (hint: start with the formulas for the components based on the angle & and then use (look up if necessary) co-function identities linking cosine and sine of 8 to sine and cosine of 0 since 8 = 3π/2-8 (this will switch cosine and sine around and eliminate - signs as well)) - C=Ccose C₁=Csine b) Prove the following formulas for the components of C: C=Ccosp1 C=Ccosp2
Notation matters when working with vectors! In particular, it is important to distinguish between the vector itself (A) and its magnitude (A). Illustrate in four separate sketches that each of the following statements is possible: a) both R = A + B and R=A+B are correct b) R = A + B is correct, but R=A+B is incorrect c) R = A + B is incorrect, but R=A+B is correct d) both R = A + B and R=A+B are incorrect
You know from your math courses that an infinitesimal segment of a circular arc can be considered as a straight line segment. Imagine that you cover a full circle in, say, the clockwise direction, with infinitesimal displacement vectors dr. Then evaluate fdr and fdr (the circle symbol on the integral just reminds us that we have to go around the full circle).
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