Download Ch 3: Motion in 2 and 3-D 3-1 The Displacement Vector DEF

Ch 3: Motion in 2 and 3-D
3-1 The Displacement Vector
DEF: Displacement Vector = the straight-line vector (single-headed arrow) representing
the length, direction, and angle for the distance between the starting and ending positions
of an object.
NOTE: Directions are expressed as N, S, E, W, or NE, (meaning literally North of East,
etc), NW, SE, or SW.
Vectors should be drawn in proportion to each other, showing relative magnitudes.
Proper notation:
→
→
→
The magnitude, or “bigness” of vector A can be written as either │A│ or ║A║, or even
A (not in bold face), and has dimensions of length.
NOTE: The magnitude of a vector is never negative!!
they use absolute value bars? ☺
Hmmm…maybe that’s why
Vector Addition There are 3 methods for vector addition:
1. Tip-To-Tail (TTT) Method – the vectors are arranged in TTT fashion, maintaining
original lengths, and relative directions and angles. The resultant vector, cleverly
called the resultant, then goes from the tail of the first to the tip of the last.
2. Parallelogram Method – two at a time, the vectors are arranged with their tails
touching at some arbitrary point (usually the origin). These two vectors then
constitute the sides of a parallelogram. The resultant then goes from their mutual
tail point, or origin, through the parallelogram, to the opposite corner. IF you have
more than two vectors, you may start by pairing them up two at a time, and use
those resultants to form a new parallelogram, and so on. You can even use it for an
odd number of vectors, saving the odd-numbered vector to be paired with the final
resultant, and so on.
3. Table Method – each of the vectors is resolved into its components along the
(arbitrarily) set x and y axes, and a table is constructed to show each respective
resolution. Sign matters!! The sum total of all of each column is configured.
The magnitude of the resultant is then computed using the Pythagorean Theorem.
Once the lengths are known, any of the inverse trig functions can be used to discern
the final angle. When it comes to specifics and enumeration, this method is the most
efficacious.
Appropriately used, the T.T.T. and
Methods will give a fairly accurate prediction
of the quadrant for the resultant, just not necessarily the magnitude and angle. If you
want accuracy on all counts, your best bet comes from using the
Method. ☺
Figure 3.1 below shows the path of a sailboat as it tacks back and forth on its journey
to a southeastern port.
Figure 3-1
Figure 3-2
Figure 3-3
Which method is being used to determine the resultant displacement in Figure 3-1? In
Figure 3-2? Figure 3-3?
See Ex 3-1, p 54. You walk 3 km E and then 4 km N. What is your resultant
displacement? Include direction and θ.
3-2 General Properties of Vectors
Figure 3-5
Q: What are the similarities and differences of the vectors drawn in Figure 3-5 above?
Actually, there are NO differences, save for how you initially placed them on the paper.
Multiplying a Vector by a Scalar
A scalar is just a number. →
→
B = │s│A
positive, and antiparallel if s is negative.
→
has magnitude │s│A, and is ║ A if s is
Subtracting Vectors
→
→
→ →
Subtracting B from A is the same as adding – B to A.
→
→
NOTE: An equivalent way of subtracting B from A is to place them tail-to-tail, noting
→
→
→
that the resultant, C, must be added to B to obtain the resultant vector A, as shown
below in Figure 3-6 (a) and (b).
Components of Vectors
The component of a vector in a given direction is a number whose absolute value is the
length of the projection of the vector onto an axis in that (arbitrarily set) direction. It’s
found by dropping a ⊥ from the head of the vector to the axis.
The projection is positive if the head of the projection is in the
the
positive direction relative to the tail. See Figure 3-7 below. Make sure you
are cognizant of quadrants when you assign the respective signs. ☺
Figure 3-7 (a)
(b)
From unit circle geometry, and trig functions:
The basic form of a circle, centered at the origin is given in:
Rectangular coordinates as
x 2 + y 2 = r2
Polar coordinates as
cos2θ + sin2θ = 1
Based on some simple manipulations that you will address later on:
x = r cos θ
and
y = r sin θ
Further manipulations of these equations, as well as the Pythagorean Theorem (P.T.),
can get you to be able to solve for most resultant vectors. But what about if you don’t
have nice, pretty right triangles?
Figure 3-9 below becomes, as they say, mui importante!
Figure 3-9
What’s happening is that vector A is being interpreted as the hypotenuse of one right
triangle, vector B is being interpreted as the hypotenuse a second right triangle, and
vector C as the hypotenuse of a third. Vector C is also the resultant for the sum of
vectors A and B. So, to solve for vector C, you need to think of it this way:
• Cx = Ax + Bx
• Cy = Ay + By
• │C│ = √(Cx2 + Cy2) and is thus solvable using the P.T.
You NEED the info above to do Example 3-2 on p 57, followed by alt int <’s are ≡ to
find the angle. Or, you can use Law of Cosines. Followed by Law of Sines to find the
angle. I’ll show you how. ☺
See Example 3-2, p 57, and Figure 3-11 below.
Unit Vectors
DEF: Unit vector = a dimensionless vector with unit magnitude. Unit vectors are often
∧
written with an overhead caret, j . Unit vectors that point in the x, y, and z directions
∧ ∧
∧
are usually written as i, j, and k, respectively. Any general vector can be rewritten as
the sum of its components. In Figure 3-12 below, you can rewrite vector A as the sum
of its x, y, and z components:
∧
∧
∧
A = Axi + Ayj + Azk
Figure 3-12
Some general rules are shown in Table 3-1 below:
Try the following as a review of resolution of vectors and / or free body diagrams:
• Ch 4 (N3): Resolution of Vectors practice sheet
• www.glenbrook.k12.il.us/gbssci/phys/shwave/fbd.html - sign in as a guest, and
complete all 12 problems. Make a mini sketch once you are right!
• Ch 4 – Morr Practice with T.T.T.,
, and
Methods
White boarding time! Exercises p 56; p 59; p 79, #38;
Grp 1
Grp 2
Grp 3
p 79, #41
Grp 4