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5 Things to remember (when doing Physics)
• Learn the principles & concepts, and master the techniques. Final answer does not matter.
• Go step by step. Do not skip steps to go to final answer.
• Learn to explain the variables in words.
• Use the drawing, before using the formula, to solve a physical problem.
• Minus signs and units are important.
Component Date Weighing Minimum Level
Assignment 1 Week 5 5% 50%
Class Test 1 Week 7 10% 50%
Assignment 2 Week 11 5% 50%
Class Test 2 Week 13 10% 50%
Practical / Project (from Lab work)
TBA 20% 50%
Final Exam TBA 50% 50%
Assessment
Assignments/ Homework
• It is in your best interest to understand the assignment/homework.
• If you have difficulty, let us discuss this at tutorial
• Please do not copy your assignment/homework. There will be no one to copy from on a test/exam.
• Collaboration is better than copying.
• It is in your best interest to help your classmates – No one really knows a subject until they explain it to
someone else.
Engineering Physics BEN503
Measurement
What is Physics?
• Physics is a science that deals with matter and energy and their interactions.
– That means, it is the study of the basic components of the universe and their interactions.
• Physics is subdivided into kinematics, mechanics, thermodynamics, optics, acoustics, etc.
• Theories of physics have to be verified by the experimental measurements.
Measurement
• A scientific measurement requires: – the definition of the physical quantity
– the units
– The value of a physical quantity is actually the product of a number and a unit. Example: 20 kg
• A unit is a measure of a quantity. – For example, 1 meter (m) is a unit of length. Any
other length can be expressed in terms of 1 meter.
International System of Units
• The SI system, or the International System of Units, is also called the metric system.
• It is the internationally accepted system of units for measurement in all of the sciences, including Physics
• The SI consists of base units and derived units: – Three basic quantities in the study of motion are the
length (L), time (T), and mass (M).
– The derived units can be expressed in terms of the base units. In other words, many units are derived from this set
– For example, speed, which is distance divided by time
Base or Fundamental Units (International System of Units)
Quantity Unit Name Symbol
Length (Distance)
meter m
Mass
kilogram kg
Time
second s
Electric current
ampere A
Temperature
kelvin K
Intensity of light
candela cd
Amount of substance
mole
mol
Derived Units (and Dimensions)
• Single Fundamental Unit – Area = Length x Length [L]2
– Volume = Length x Length x Length [L] 3
• Combination of Units – Velocity = Length / Time [L/T]
– Acceleration = Length / (Time x Time) [L/T2]
– Force = Mass x Length / (Time x Time) [M L/T2]
– Density = Mass / (Length x Length x Length) [M/T3]
Physical Quantities
• Must always have dimensions
• Can only compare quantities with the same dimensions
v = v(0) + a x t
[L]/[T] = [L]/[T] + [L]/[T]2 [T]
• Comparing quantities with different dimensions is nonsense
v = a x t2
[L]/[T] = [L]/[T]2 [T]2 = [L]
Which of these could be correct?
Provides Solutions sometimes
• Period of a Pendulum
m
le
g
mld
g
lc
gmlb
gla
)
)
)
)
)
2
2
g
l 2
Period is a time [T] -
Can only depend on:
Length [L] - l
Mass [M] - m
Gravity [L/T2] - g
Length
• Redefining the meter:
• In 1792 the unit of length, the meter, was defined as one-millionth the distance from the north pole to the equator.
• Later, the meter was defined as the distance between two finely engraved lines near the ends of a standard platinum-iridium bar, the standard meter bar. This bar is placed in the International Bureau of Weights and Measures near Paris, France.
• In 1983, the meter was defined as the distance traveled by light in a vacuum during the time interval of 1/299 792 458 of a second. The speed of light is then exactly 299 792 458 m/s.
Time
• Atomic clocks give very precise time measurements.
• An atomic clock at the National Institute of Standards and Technology in Boulder, CO, is the standard.
• In 1967 the standard second was defined to be the time taken by 9 192 631 770 oscillations of the light emitted by cesium-133 atom.
Mass
• A platinum-iridium cylinder (height = diameter = 39 mm), kept at the International Bureau of Weights and Measures near Paris, has the standard mass of 1 kg.
• Another unit of mass is used for atomic mass measurements.
• Carbon-12 atom has a mass of 12 atomic mass units, defined as
kgxu 271086538660.11
Changing Units
• Sometimes, we may need to change the units of a given quantity (using the chain-link conversion). – For example, since there are 60 seconds in 1 minute,
• Conversion between one system of units and another can be made.
• The first equation above is called the “Conversion Factor”, which is defined as a ratio of units that is equal to 1
ss
s
s
120)min1
60(xmin)2()1(xmin)2(min2
and,min1
601
60
min1
Significant Figures (Rules for deciding the number of significant figures in a measured quantity)
• All nonzero digits are significant – 1.234 g has 4 significant figures – 1.2 g has 2 significant figures.
• Zeroes between nonzero digits are significant – 1002 kg has 4 significant figures – 3.07 mL has 3 significant figures.
• Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: – 0.001 oC has only 1 significant figure – 0.012 g has 2 significant figures.
• Trailing zeroes that are also to the right of a decimal point in a number are significant – 0.0230 mL has 3 significant figures – 0.20 g has 2 significant figures.
Significant Figures (Rules for deciding the number of significant figures in a measured quantity)
• When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: – 190 miles may be 2 or 3 significant figures – 50,600 calories may be 3, 4, or 5 significant figures.
• The potential ambiguity in the last rule can be avoided by the use of
standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as: – 5.06 × 104 calories (3 significant figures) – 5.060 × 104 calories (4 significant figures) – 5.0600 × 104 calories (5 significant figures).
• By writing a number in scientific notation, the number of significant
figures is clearly indicated by the number of numerical figures in the 'digit' term as shown by these examples. This approach is a reasonable convention to follow.
Scientific Notation
Factor Prefix Symbol Factor Prefix Symbol
103 kilo k 10-3 milli m
106 mega M 10-6 micro μ
109 giga G 10-9 nano n
• All Physics quantities should be written as scientific notation, which uses the power of 10. – Example: 3 560 000 000 m = 3.56 x 109m.
• The Order of magnitude of a number is the power of ten when the number is expressed in scientific notation
• Sometimes special names are used to describe very large or very small quantities. – For example, 2.35 x 10-9 seconds = 2.35 nanoseconds (ns)
Engineering Physics BEN503
Vectors & Scalars
Vectors & Scalars
VECTORS • Quantities which indicate both magnitude and
direction • Examples: displacement, velocity, acceleration,
force, torque – Displacement Vector is a change in position. It is calculated as the final
position minus the initial position.
SCALARS • Quantities which indicate only magnitude • Examples: Time, mass, density, speed, temperature,
distance, electric charge
Vectors & Scalars
• Arrows are used to represent vectors. – The length of the arrow signifies magnitude
– The head of the arrow signifies direction
• Sometimes the vectors are represented by bold lettering, such as vector a. Sometimes they are represented with arrows on the top, such as
• Figure (a) All three arrows have the same magnitude and direction and thus represent the same displacement; (b) All three paths connecting the two points correspond to the same displacement vector
a
Some Vector Properties • Two vectors that have the same direction are said to be
parallel. • Two vectors that have opposite directions are said to be anti-
parallel. • Two vectors that have the same length and the same direction
are said to be equal no matter where they are located. • The negative of a vector is a vector with the same magnitude
(size) but opposite direction
Magnitude of a Vector
• The magnitude of a vector is a positive number (with units!) that describes its size.
• Example: – The magnitude of a displacement vector is its length.
– The magnitude of a velocity vector is often called speed.
• The magnitude of a vector is expressed using the same letter as the vector but without the arrow on top of it.
AAAofMagnitude
)(
Adding Vectors geometrically
• Vector a and vector b can be added geometrically to yield the resultant vector sum, s. – Example, double displacement of particle
• Place the second vector, b, with its tail touching the head of the first vector, a. The vector sum, s, is the vector joining the tail of a to the head of b.
Some rules for Vectors
• Commutative law (the order of vector addition doesn’t matter)
• Associative law
• Subtraction
a b b a
( ) ( )a b c a b c
( )d a b a b
Vector Addition
CDCBAR
EACBAR
Components of Vectors
• The projection of the vector on an axis is called its component.
• The process of finding the components of a vector is called resolution of the vector. Resolve vectors into x and y components; add components
• The components form the legs of a right triangle whose hypotenuse is the magnitude of the vector
cosxa a
sinya a
2 2
x ya a a Magnitude: Direction: 1tany
x
a
a
Measuring Vector Angles
Θ = 0
Θ = 90
Θ = 180
Θ = 270
x
y
90 < Θ < 180
cos (-) sin (+)
180 < Θ < 270
cos (-) sin (-)
0 < Θ < 90
cos (+) sin (+)
270 < Θ < 360
cos (+) sin (-)
• You can also use sine and cosine curves to find the signs at different quadrants
• The equations for cosθ, sinθ and tanθ in the previous slide are valid only if the angle is measured from the positive direction of the x-axis.
Unit Vectors
• Unit vectors provide a convenient means of notation to allow one to express a vector in terms of its components
• Unit vectors have magnitudes of unity and are directed in the positive directions of the x, y, and z axes, respectively, in a right-handed coordinate system.
• Therefore vector, , with components ax and ay in the x- and y-directions, can be written in terms of the following vector sum:
a
jaiaayx
ˆˆ
The unit vectors are dimensionless vectors that point in the direction along a coordinate axis that is chosen to be positive
kji ˆ,ˆ,ˆ
Addition of Vectors (by means of Components)
C A B
1tan ( / )y xC C
Where cos and cosx A y AA A A A
cos and sinx B y BB B B B
Then Cx = Ax + Bx and Cy = Ay + By
The components Ax and Ay of a vector A are numbers; they are not vectors !
Multiplying Vector by scalar
• Multiplying a vector by a scalar changes the magnitude but not the direction:
vssxv
Multiplying a vector by a vector (Scalar or dot product) • The scalar product between two vectors is
written as:
• It is defined as:
• Here, a and b are the magnitudes of vectors a and b respectively, and f is the angle between the two vectors.
• The dot product is a scalar.
• If the angle between two vectors is 0°, dot product is maximum; if the angle between two vectors is 90°, dot product is zero
Multiplying a vector by a vector (vector or cross product) • The vector product between two vectors a
and b can be written as:
• It can be proven that the magnitude of:
• Here, a and b are the magnitudes of vectors a and b respectively, and f is the angle between a and b vectors.
• The cross product is a vector.
• Direction is determined by right-hand rule
c a b
| a b |= a b sin f
Cross product
• By definition, the cross product of these vectors (pronounced “v1 cross v2”) is given by the following determinant.
• Note that the cross product of two vectors is another vector!
• Cross products are used a lot in physics, e.g., torque is a vector defined as the cross product of a displacement vector and a force vector. We’ll learn about torque in another unit.
Let v1 = x1, y1, z1 and v2 = x2, y2, z2 .
v1 v2 = x1 y1 z1
x2 y2 z2
i j k
= (y1 z2 - y2 z1) i - (x1 z2 - x2 z1) j + (x1 y2 - x2 y1) k
Dot product vs Cross Product
• The dot product of two vectors is a scalar; the cross product is another vector (perpendicular to each of the original).
• A dot product is commutative; a cross product is not. In fact,
• Dot product definition:
• Vector product definition
•
a b = - (b a)
x1, y1, z1 x2, y2, z2 = x1 x2 + y1 y2 + z1 z2
v1 v2 = x1 y1 z1
x2 y2 z2
i j k
| a b |= a b sin f
Check your understanding
• Two vectors, A and B, have vector components that are shown (to the same scale) in the first row of drawings. Which vector R in the second row of drawings is the vector sum of A and B?
Check your understanding (The Component Method of Vector Addition)
• A jogger runs 145 m in a direction 20.0° east of north (displacement vector A) and then 105 m in a direction 35.0° south of east (displacement vector B).
• Determine the magnitude and direction of the resultant vector C for these two displacements
Addition of Vectors (by means of Components) - Example
Problem Solving Strategy
IDENTIFY AND SET UP • Target variable: vector
magnitude, its direction or both
• Draw individual vectors and coordinate axes
• Tail of 1st vector in origin, tail of 2nd vector at the head of 1st vector, and so on…
• Draw the vector sum from the tail of 1st vector to the head of the last vector.
• Make a rough estimate of magnitudes and direction.
EXECUTE • Find x- and y-components of
each individual vector
• Check quadrant sign! • Add individual components
algebraically to find components of the sum vector
• Magnitude
• Direction
Bx BB cos
By BB sin
... xxxx CBAR... yyyy CBAR
)arctan(x
y
R
R
22 )()( yx RRR
EVALUATE Check your results comparing them with the rough estimates!
Addition of Vectors (by means of Components) - Example
• θA=90.0-32.0=58.0
• θB=180.0+36.0=216.0
• θC=270.0
• Ax=A cos θA
• Ay=A sin θA
Distance Angle X-comp Y-comp
A=72.4m 58.0 38.37m 61.40m
B=57.3m 216.0 -46.36m -33.68m
C=17.8m 270.0 0.00m -17.80m
-7.99m 9.92m
12999.7
92.9arctan
7.12)92.9()99.7( 22
mm
mmmR
Example, vectors:
