Download Units

1. MEASUREMENTS AND VECTORS
Units:
There are 3 fundamental quantities- length, mass and time.
• SI (System International) Units:
– mks: Length = meters (m), Mass = kilograms (kg),
Time = seconds (s)
• British Units:
– Inches, feet, miles, pounds
Other system of units:
– cgs: L = centimeters (cm), M = grams (gm), T =
seconds (s)
Conversion of units:
• Useful Conversion factors:
–
–
–
–
1 inch
1m
1 mile
1 mile
=
=
=
=
2.54
3.28
5280
1.61
cm
ft
ft
km
• Example: convert miles per hour to meters per
second:
1
mi
mi
ft
1 m
1 hr
m
1
 5280


 0.447
hr
hr
mi 3.28 ft 3600 s
s
DIMENSIONS OF PHYSICAL QUANTITIES:
length(L) , mass (M), time(T)
• Example:
Doing a problem you get the
answer distance
t= dv (velocity x time2)
Units on left side = T
Units on right side = L x L / T
= L2 / T
Left units and right units don’t
match, so answer must be
wrong!!
• Scalars : Quantities having magnitude only. Example: Mass,
Distance, Speed.
• Vectors: Quantities having both magnitude and direction.
Example: Displacement, Force, Velocity, Acceleration
• Distance : Total length covered by an object.
• Displacement : The shortest distance between any 2 points in
a specified direction.
• Distance and displacement have the same units. (m, cm, km)
Unit Vectors:
• A Unit Vector is a vector having length 1
and no units
• It is used to specify a direction
• Unit vector u points in the direction of U
– Often denoted with a “hat”: u = û
U
û
y
• Useful examples are the Cartesian
unit vectors [ i, j, k ]
– point in the direction of the
x, y and z axes
j
k
z
i
x
Vector Notation:
• There are two common ways of
indicating that something is a vector
quantity:
– Boldface notation: A

A
– “Arrow” notation:

A= A
Vectors...
• The components of r are its (x,y,z)
coordinates
– r = (rx ,ry ,rz ) = (x,y,z)
where r = |r |
• Consider this in 2-D (since it’s easier to
(x,y)
y
draw):
– rx = x = r cos 
– ry = y = r sin 
arctan( y / x )
r

x
Magnitude of a vector:
•
The magnitude (length) of r is found using the Pythagorean theorem:
r
y
r  r  x2  y 2
x

The length of a vector clearly does not depend on its direction.
Vector addition using components:
A= (Ax i + Ay j+ Az k)
What is the resultant vector, Z, from
adding A+B+C?
Vector A = (1,2,3)
Vector B = (4,5,0)
Vector C = (2,-3,2)
Z = (AXi + AYj + AZk) + (BXi + BYj + BZk) + (CXi + CYj + CZk)
= (AX + BX + CX)i + (AY + BY+ CY)j + (AZ + BZ + CZ)k
= (1 + 4 + 2)i + (2 + 5 - 3)j + (3+ 0 + 2)k
= {7,4,5}
Adding Vectors by Components
Any vector can be expressed as the sum of two
other vectors, which are called its components.
Usually the other vectors are chosen so that
they are perpendicular to each other.
3-4 Adding Vectors by Components
If the components are perpendicular,
they can be found using
trigonometric functions.
Adding Vectors by Components
The components are effectively one-dimensional, so they
can be added arithmetically:
Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use: