Physics 1116 Fall 2011 Homework 1 Due: Wednesday, August 31st

Reading: Kleppner & Kolenkow Chapter 1.1-1.8 and Note 1.1

1. Vector Algebra. Consider two vectors a = − i + 2 j + 3 k and B = 4 i − 6 j + 3 k.Compute:

a. a + B

b. a− 2B

c. a ·Bd. a×B

e. B×a

f. a · (a×B) and B · (a×B)

2. Distance of closest approach. Consider an arbitrary point in space given by a vectorr0, and an arbitrary direction given by a unit vector n. We can define a straight linein space, passing through the given point in the given direction, by the locus of pointsr = r0 + ηn for all values of a real number η. Use vector techniques to answer thefollowing questions.

a. What is the distance of closest approach of this line to the origin?

b. If a new line is drawn from the origin to the point of closest approach, at whatangle do these two lines intersect one another?

c. List two possible conditions that, if either is true, the distance of closest approachwill be zero.

Comment. In experimental particle physics it is often important to know the distanceof closest approach of a particle track to the point where the primary beams collided.In many cases this distance helps to measure some other particle’s lifetime.

3. Kinematics in one dimension with contant acceleration. KK Problem 1.13

4. Kinematics in two dimensions with contant acceleration. KK Problem 1.21

(You might want to work out the problem first for the traditional case of throwing therock while standing on level ground.)

5. Kinematics of Uniform Circular Motion. The earth has a diameter of approxi-mately d = 1.3×107 m; a day is about 86,400 seconds; and the latitude of Ithaca isabout 42◦ N.

a. What is the angular velocity ω of a tree (a) at the equator, (b) in Ithaca?

b. What is the magnitude of the velocity (relative to a non-rotating earth) of a tree(a) at the equator, (b) in Ithaca?

c. What is the direction of the velocity vector of a tree (a) at the equator, (b) inIthaca?

d. What is the magnitude of the acceleration (relative to a non-rotating earth) of atree (a) at the equator, (b) in Ithaca?

e. What is the direction of the acceleration vector of a tree (a) at the equator, (b) inIthaca?

6. More Kinematics of Uniform Circular Motion. A wheel of radius R has a dotpainted on the rim at the position r = R, φ = 0. The wheel begins to roll to the left(without slipping) at constant angular velocity ω, so the dot traces out a cycloid curve.We will choose ω > 0 so the wheel moves leftward as t increases. See Figure 2, below,for the geometry.

a. Find r(t). Hint: decompose this into the sum of two vectors, one locating the centerof the wheel with respect to the ground, and one locating the dot with respect tothe center of the wheel.

b. Find v(t).

c. Find a(t).

d. At the instant that the dot is at the top of the wheel, what is the radius of curvatureof its path? (The radius of curvature is the radius of the circle that matches uplocally with the path at a given point.) Hint: you know v(t) and a(t).

φφ=0

y

x

Figure 1: Configuration for Problem 6. The wheel initially was in the position where the dotwas at the φ = 0 axis. This snapshot shows the wheel shortly after rolling began, ie afterφ has rotated about 40◦. The small black arrow indicates the direction of rotation. The xycoordinate system is indicated by the red vectors.

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