7. In the diagram below, the vectors
and 
are
at right angles to each other. The length of 
is c and the length of is d. The horizontal
and vertical components of 
are a and b
respectively.
The vertical component of , e, is
A. a d / c.
B. b d / c.
C. b d / a.
E. None of the above.
ANSWER
A is the correct answer. In order to solve this problem, it helps to add two
named angles to
the diagram:
We are looking for e, so we need sin of 
. We
note that θ is the complementary angle of ,
so sin() = cos(θ) = a/c . The final answer is
d sin() = a d / c.
RELEVANCE TO PHYSICS 1
This question requires math skills you will need to solve inclined plane
problems and to
calculate vector directions for gravitational and electric forces.
8. Consider two vectors,
and 
,
and their sum, 
=
+ 
.
Which orientation of the
vectors shown below gives the smallest magnitude (length) of 
? (The magnitudes of 
and 
don’t change in the diagrams , but don’t
as sume that they are drawn to scale .)
D. It doesn’t make any difference since the magnitudes of
and
don’t change.
E. There is no way to tell without knowing the magnitudes of
and .
ANSWER
A is the correct answer.
RELEVANCE TO PHYSICS 1
We use vectors a lot in Physics 1. This question is trying to get you to
think about how
vectors are aligned to give the minimum sum: opposite directions. The alignment
of vectors
is critical when we consider speeding up / slowing down, centripetal
accele ration , torque,
work, and many other topics.
9. (Challenge problem.) The vectors
and are
given as shown below, with lengths c and d
respectively. The angle of with respect to
the reference line is θ and the angle of
with
respect to the reference line is , going
counter-clockwise
The magnitude (length) of the sum, =
+ , is
E. The magnitude of 
cannot be determined
unless the direction of the reference line is
known.
ANSWER
D is the correct answer. The easiest way to get this result is to realize
that the magnitude of a
vector sum is independent of the choice of coordinate systems . In that case,
pick the X axis
to align with the u vector. The angle of the v vector with respect to the X axis
is (-θ). We
take the X and Y components of v using trigonometry , add the X component of u,
and finally
use the Pythagorean Theorem to get the length of the sum. Just for laughs, we
put cos(θ-)
instead of cos(-θ) because cos(–α) = cos(+α).
Note that (θ-) and (-θ)
are independent
of the angle of the reference line.
While we’re on the subject, let’s go over how to convert
vectors from (Length, Angle)
coordinates to (X, Y) coordinates and back. First, assume that the angle has been
measured in
the counter-clockwise direction from the +X axis. This is the standard you will
see most
often and the one we will use in Physics 1. To convert from (Length, Angle) to (X,Y),
we use
the following formulas (Let length = L and angle = α.):
X = L cos(α)
Y = L sin(α)
Converting from (X, Y) to (Length, Angle) is a little
trickier. We have to be careful to get the
angle in the proper quadrant. First, the easy part. The length is given by the
Pythagorean
Theorem:
Calculation of the angle depends on the quadrant we are
in. First, the special cases. If X = 0
and Y = 0, then by convention we say L = 0 and α = 0. If X = 0 and Y > 0, then α
= π/2 in
radians or 90º. If X = 0 and Y < 0, then α = –π/2 in radians or –90º. If Y = 0
and X > 0, then
α = 0. If Y = 0 and X < 0, then α = π in radians or 180º
In cases where X > 0, we are in quadrants I or IV. Use
this formula:
In cases where X < 0 and Y > 0, we are in quadrant II. Use
this formula:
In cases where X < 0 and Y < 0, we are in quadrant III.
Use this formula:
Following these rules , a will always be in the range (–π,
+π] or (–180º,+180º]. On
computers, you will find these formulas programmed into one function:
atan2(Y,X).
RELEVANCE TO PHYSICS 1
This question is more complex than most vector calculations we will need in
Physics 1, but it
brings out some important concepts that you will use in later courses at RPI. In
any case,
make sure you understand how to convert a vector from (Length, Angle) form to (X, Y)
form
and back.
10. Dick and Jane went into business selling mud pies.
They set up two mud pie stands at
opposite ends of the neighborhood to maximize their potential customers. On
Monday, they
sold a total of 100 pies. One Tuesday, Dick took the day off because his sales
were so good
on Monday. Jane figured that she could triple her sales from Monday if she
copied Dick’s
slick techniques. Unfortunately, Dick’s customers from Monday all returned their
pies to
Jane on Tuesday. Jane met her Tuesday sales target, but she only sold 20 new
pies after
reselling all of Dick’s returned pies from Monday. What equations would you use
to
determine how many pies Jane sold on Monday?
A.
J + D = 100
3J = 120
B.
J + D = 100
3J – D = 20
C.
J + D = 100
3J + D = 120
D. There is not enough information to determine a unique answer.
E. There is contradictory information so there is no answer.
ANSWER
B is the correct answer. The second equation could also have been written as
3J = D+20.
RELEVANCE TO PHYSICS 1
We often get pairs of linear equations in Physics 1 when we solve problems
using Newton’s
Second Law . Many problems involve objects connected by ropes. A variable
re presenting
tension in a rope normally appears in two equations, one for each end of the
rope. With the
right choice of coordinate systems, the tension term will be + in one equation
and – in the
other. The easiest way to solve a system of equations like that is simply to add
the two
equations. (Easy = less likely to make an error.)
To solve the two equations in B, add them together to get
J + 3J + D – D = 100+20
Solve for J, then substitute J back into either original equation to solve for
D.
