Physics 1210: Study Guide for Chapter 1
OBJECTIVES:
1. Learn the basic quantities and their SI units (metric system).
2. Understand how to determine significant figures.
3. Know how to convert units.
4. Review coordinate systems.
5. Understand the difference between scalars and vectors.
6. Review unit vector notation.
7. Learn to use graphical and trigonometric methods to find the resultant of several
vectors.
The basic physical quantities are length (L), mass (M) and time (T).
The SI units are:
Meter (m) for length
Kilogram (kg) for mass
Second (s) for time
Significant figures (useful digits) – also known as sig. figs.
Multiplication/division: Limited by the number with the least sig. figs.
With multiplication and division, one should be concerned with the
number of significant digits. In multiplying (dividing) two or more
quantities, the number of significant figures in the final answer should be
equal to the least number of significant figures in the multiplicand
(dividends). Do NOT round intermediate numbers, but only the final
answer. Never truncate numbers; that is, never chop off the unwanted
digits without rounding.
Addition/subtraction: Limited by the digit place with respect to the decimal.
In addition and subtraction one needs to pay attention to the number of
decimal places and not the number of significant digits. In this case, the
number of decimal places in the result should equal the smallest number of
decimal places of any term in the sum (or difference).
Conversion of Units – Think: multiply by one.
Consider the units you want to change. Find the relationship between the new and
the old units. Set up a ratio such that the unit you want to cancel is opposite (top
or bottom) the unit in the original number. This way, when you multiply by the
ratio you are multiplying by one and will only have the effect of changing units.
Example –
m  1km  60 s  60 min 
3 km
Speed of sound: vs = 331 


 = 1.19 ×10
s  1000 m  1 min  1hr 
hr
If you want to change a unit with a power (that is squared, cubed, …), you have to
change the linear (no power) unit first, then raise the result to the correct power
(square, cube, …).
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Example –
Density of air: ρ air = 1.2
3
kg  1000g  1m 
−3 g
 = 1.2 ×10

3
m  1kg  100cm 
cm 3
Know the prefixes for powers of ten.
COORDINATE SYSTEMS:
Number line:
A line marked by scale, going from negative infinity to positive infinity.
Cartesian Coordinate System:
A Cartesian coordinate system consists of two perpendicular number lines,
representing two different directions (for 2-Dimensional space). [For 3-dimensional
space, you will need three perpendicular number lines.] Points in this space can be
located with two numbers; one for each number line (axis). In the case of the figure
below, we have an x-axis (horizontal) and a y-axis (vertical). Points in this coordinate
system can be located with (x, y) pairs, where x = 0 and y = 0 is where the axes cross. We
call this the origin. Point A in the figure is located at (3, 2), while point B is located at (1,
-2). Notice that the coordinate system is broken in to four quadrants (I, II, III and IV).
Point A is in quadrant I, while point B is in quadrant IV.
Quadrants –
I: where x > 0 and y > 0
II: where x < 0 and y > 0
III: where x < 0 and y < 0
IV: where x > 0 and y < 0
Polar Coordinate System:
A polar coordinate system uses the same axes as
the Cartesian coordinate system, but instead of using (x,
y) pairs for the location of a point it uses (r, q) pairs.
Here, the r represents the radial coordinate. This is the
distance from the origin and is always positive. The
other coordinate, q, is the angular coordinate. This
represents the angle, counter clockwise from the
positive x-axis.
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VECTORS AND SCALARS:
A value of a scalar can be completely defined by only one number. The value of a vector
requires two numbers in 2-dimensions and three numbers in 3-dimensions. Two vectors
are identical iff (if and only if) they have the same magnitude and the same direction.
   
Addition of vectors: R = A + B + C +...
Graphical method:
A vector can be depicted by an arrow. The length of the arrow represents the magnitude
of the vector. The direction is represented by the angle the arrow makes with the positive
x-axis. Draw the first vector according to the scale chosen such that the length represents
the magnitude and the direction is given by the angle. To add vectors graphically, place
the tail of the second vector at the tip of the first vector using the appropriate length and
orientation for the second vector. Continue adding vectors in this tip to tail fashion until
the last vector given is completed. Draw an arrow from the tail of the first vector to the
tip of the last vector. This vector is the resultant, R, of the sum of all the vectors given.
Also note that the order in which the vectors added does not affect the value of the
resultant. In the example below three vectors are added. The resultant vector ends up
pointing into quadrant II.
Component Method:
First, resolve the vectors into components. If the vector is along the x-axis then the ycomponent is zero. If the vector is along the y-axis then the x-component is zero.
Remember that the components can be negative. To find the resultant vector, you must
add the x- and y-components separately.
•
•
Add all the x - components. Rx = Ax + Bx+ ….
Add all the y - components. Ry= Ay + By + ….
•
Apply the Pythagorean theorem to find the resultant magnitude. R = Rx2 + Ry2
•
Use Rx & Ry to find the direction by tan qR = opp/adj.
The direction can be represented with respect to the + x-axis or with respect to the
directions north (N), south (S), east (E) and west (W). Example: 25° N of E. To find this
direction, point in the direction of the second direction given (E) and then angle toward
the first direction given (N). In this example, the vector is in the first quadrant.
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Example (Addition of Vectors):
Find the resultant of the following three vectors.
qA = 30° N of E
A = 5.00 km
qB = 50° N of W
B = 10.0 km
qC = 20° E of S
C = 20.0 km
Resolve the vectors into x and y components.
Vector A:
30o
Ax
Vector B:
By
50
Bx
o
Vector C:
Cy 20o
Ay
Ax = A cos q = 5.00 km cos 30° = 4.33 km
Ay = A sin q = 5.00 km sin 30° = 2.50 km
Bx = – B cos q = –10.0 km cos 50° = – 6.43 km
By = B sin q = 10.0 km sin 50° = 7.66 km
Cx = C sin q = 20.0 km sin 20° = 6.84 km
Cy = – C cos q = –20.0 km cos 20° = –18.79 km
(Note that in finding the x and y components of C, sine and
cosine are reversed. Please make sure to always look
opposite and adjacent when working with sine and cosine.)
Cx
After
resolving the vectors that need to be resolved, then add the xcomponents and y-components separately. This will give you
the x- and y-components of the resultant vector.
Rx = 4.33 – 6.43 + 6.84 = 4.74 km
Ry = 2.50 + 7.66 + -18.79 = – 8.63 km
R = Rx2 + Ry2 = (4.74km)2 + (−8.63km)2 = 9.85km
We can find the angle of the resultant vector by using tanq = opp/adj. This resultant
vector points in the positive x-direction and the negative y-direction; that is, down and to
the right. So to get an angle that is S of E, we will use |Ry| as the opposite side and |Rx| as
the adjacent side.
tan θ R =
Ry
Rx
è
 | R |
 | −8.63km | 

θ R = arctan  y  = arctan 
 = 61.2
 | 4.74km | 
 | Rx | 
S of E
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-61.2
o
This angle can also be written with respect to the positive x-axis. Because
the angle is counter clockwise from the + x-axis, it can be written as qR =
-61.2°.
PRACTICE MULTIPLE-CHOICE QUESTIONS:
1. The prefix micro represents
a) 1/10
c) 1/1000
b) 1/100
d) 1/1,000,000
2. A centimeter is
a) 0.001 m
b) 0.01 m
For the following, know that: 1 in = 2.54 cm
c) 0.1m
1 ft = 12 in
d) 10 m
1 mi = 5280 ft
3. Of the following, which is the shortest?
a) 1 mm
b) 0.01 in
0.00001 km
c) 0.001 ft
d)
4. Of the following, the longest is
a) 104 in
b) 104 m
mi
c) 103 ft
d) 0.1
5. A height of 5 ft 8 in. is equivalent to
a) 173 cm
b) 177 cm
cm
c) 207 cm
d) 223
6. The number of seconds in a month is approximately
a) 2.6 x 106
b) 2.6 x 107
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x10
c) 2.6 x 108
d) 2.6
7. The ratio of the prefixes Mega/milli has a value of
a) 103
b) 106
c) 109
d) 1018
8. Which expression is dimensionally equivalent t-1; where v is velocity (L/T), x is
distance (L), and t is time (T).
a) v/x
b) v2/x
c) x/t
d) v2t
9. If a is acceleration (L/T2), v is velocity, x is position, and t is time, then which
equation is dimensionally incorrect?
a) t =x/v
b) v =a/t
c) a = v2/x
d) t2 = 2x/a
10. For the equation: x = bt3 +ct4, where b and c are constants, what are the
dimensions for b and c respectively? (x is position and t is time.)
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a) T3, T4
b) T-3, T-4
c) L2T3, L2T4
d) L/T3, L/T4
11. Multiplying a 2 significant figure number by a 3 significant figure number and
then divide the product by a six significant figure number. The answer will have
how many significant figures?
a) 5/6
b) 1 c) 2 d) 11 e) 8
12. Which of the following units could be associated with a vector quantity?
a) km/s2
b) kg
c) hours
d) m3
13. The magnitude of the resultant of two forces is a minimum when the angle between them
is:
a) 0°
b) 45°
c) 90°
d) 180°
14. Which of the following pairs of displacements cannot be added as vectors to give a
resultant displacement of 2 m?
a) 1 m and 1m
b) 1 m and 2 m
c) 1 m and 3 m
d) 1 m and 4 m
15. Which of the following sets of forces cannot have a vector sum of zero?
a) 10,10, and 10 N
b) 10, 10, and 20 N
c) 10, 20 ,and 20 N
d) 10, 20, and 40 N
16. Fred walks 8 km north and then 5 km in a direction 69° east of north. His resultant
displacement from his starting point is:
a) 11 km
b) 12 km
c) 13 km
d) 14 km
17. The resultant of a 4-N force acting vertically and a 3-N force acting horizontally is:
a) 1 N
b) 5 N
c) 7 N
d) 12 N
18. The angle between the resultant in the previous question and the vertical is
approximately:
a) 37°
b) 45°
c) 53°
d) 60°
19. An airplane travels 100 km to the north and then 200 km to the east. The displacement of
the airplane from its starting point is approximately:
a) 100 km
b) 200 km
c) 220 km
d) 300 km
20. At what angle east of north should the airplane in the previous question have headed in
order to reach its destination in a straight flight?
a) 22°
b) 45°
c) 50°
d) 63°
21. Two forces of 10 N each act on an object. The angle between the forces is 120°. The
magnitude of their resultant is:
a) 10N
b) 14N
c) 17N
d) 20 N
22. A vector A lies in a plane and has the components Ax and Ay. The value of Ax is equal to:
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a) A – Ay
b)
A 2 − Ay2
c)
A − Ay
d)
A 2 − Ay2
23. An escalator has a velocity of 3.0 m/s at an angle of 60° above the horizontal. The
vertical component of its velocity is:
a) 1.5 m/s
b) 1.8 m/s
c) 2.6 m/s
d) 3.5
m/s
24. The following forces act on an object: 10 N to the north, 20 N at 45° south of east, and
5 N to the west. The magnitude of their resultant is:
a) 5 N
b) 10 N
c) 13 N
d) 23 N
ANSWER KEY TO THE MULTIPLE-CHOICE QUESTIONS:
1. D
2. B
3. B
4. B
5. A
6. A
7. C
8. A
9. B
10. D
11. C
12. A
13. D
14. D
15. D
16. A
17. B
18. A
19. C
20. D
21. A
22. D
23. C
24. B
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