MA330: Assignment 2
Required Reading.
• Read Chapter 1, § 16-18, 24-30, 32
To be turned in March 15th at the start of class.
1. Textbook, page 44, #8
2. Textbook, page 66, #2
3. Textbook, page 66, #4
4. Textbook, page 74, #1ab
5. Textbook, page 74, #3
6. Textbook, page 78, #4
7. (a) Using trigonometry, show that if a vector is rotated counterclockwise by an angle θ then the new vector is
cos θ − sin θ
x
x
=
sin θ cos θ
y old
y new
(b) Show that
1
0
0
a
stretches vectors in the y direction.
(c) Beginning with the basis {i, j} create a new basis {b1 , b2 } by first scaling, then rotating the original basis.
(d) Show how to compute dot products in this new basis.
(e) Show how to compute cross products in this new basis.
8. A particle moves from right to left along the curve y = x2 . At time t = 1, the particle is located at (1, 1), has
speed 2 and is slowing down at the rate 3. Find its acceleration vector at that time. Hint: express the curve as
r(t) = hx(t), x(t)2 i.
9. Consider a pendulum oriented so that the rest position is downward. Suppose the pendulum is made from a massless
wire of length L and a point mass m. Define r(t) to be the mass at time t.
(a) Using Newton’s laws, prove
mr̈ = −mg k −
T
r
L
where g is the acceleration of gravity and T is the (unknown) tension in the wire.
(b) Prove that r · ṙ = 0. Hint: note that krk = L
(c) By dotting the equation of motion with ṙ, show that the unknown tension can be eliminated.
m ṙ · r̈ = −mg k · ṙ
and that this can be expressed as conservation of energy
1
mkṙk2 + mgz = E = constant
2
1