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Chapter 2 : Physics & Measurement & Mathematical ReviewWeerachai Siripunvaraporn
Department of Physics, Faculty of Science
Mahidol University
email&FB : [email protected]
Physics
Fundamental Science Concerned with the fundamental principles of the
Universe Foundation of other physical sciences Has simplicity of fundamental concepts
Introduction
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Objectives of Physics
To find the limited number of fundamental laws that govern natural phenomenaTo use these laws to develop theories that can predict the results of future experimentsExpress the laws in the language of mathematics
Mathematics provides the bridge between theory and experiment.
Introduction
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Theory and Experiments
Should complement each otherWhen a discrepancy occurs, theory may be modified or new theories formulated.
A theory may apply to limited conditions. Example: Newtonian Mechanics is confined to objects
traveling slowly with respect to the speed of light. Try to develop a more general theory
Introduction
CH1
Measurements
Used to describe natural phenomenaEach measurement is associated with a physical quantityNeed defined standardsCharacteristics of standards for measurements
Readily accessible Possess some property that can be measured reliably Must yield the same results when used by anyone
anywhere Cannot change with time
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Fundamental Quantities and Their Units
Quantities Used in MechanicsIn mechanics, length, mass and time are used:
All other quantities in mechanics can be expressed in terms of the three fundamental quantities.
Feel the numbers… It is important to develop a ‘feeling’ for some of the numbers
that you use.
1 kg = 1 liter of water = 1000 cc
http://www.mathsisfun.com/measure/metric-length.html
1 m Duration of a heart beat when resting
Prefixes
Prefixes correspond to powers of 10.Each prefix has a specific name.Each prefix has a specific abbreviation.The prefixes can be used with any basic units.They are multipliers of the basic unit.Examples:
1 mm = 10-3 m 1 mg = 10-3 g
Section 1.1
CH1
Prefixes, cont.
Section 1.1
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For example,
1 nm = 10-9 m; 3 Gs = 3 x109 s;
8.9 Tm = 8.9 x 1012 m; 6 g = 6 x 10-6 g;
Dimensional Analysis
Technique to check the correctness of an equation or to assist in deriving an equationDimensions (length, mass, time, combinations) can be treated as algebraic quantities.
Add, subtract, multiply, divideBoth sides of equation must have the same dimensions.Any relationship can be correct only if the dimensions on both sides of the equation are the same.
Section 1.3
CH1
Dimensional Analysis, exampleGiven the equation: x = ½ at 2
Check dimensions on each side:
The T2’s cancel, leaving L for the dimensions of each side.
The equation is dimensionally correct. There are no dimensions for the constant.
Section 1.3
CH1
Symbols
The symbol used in an equation is not necessarily the symbol used for its dimension.
Some quantities have one symbol used consistently.
For example, time is t virtually all the time.
Some quantities have many symbols used, depending upon the specific situation.
For example, lengths may be x, y, z, r, d, h, etc.
The dimensions will be given with a capitalized, non-italic letter.
The algebraic symbol will be italicized.CH1
Conversion of UnitsWhen units are not consistent, you may need to convert to appropriate ones.Units can be treated like algebraic quantities that can cancel each other out.Always include units for every quantity, you can carry the units through the entire calculation.
Section 1.4
CH1
Exercise Convert 1 kg/m3 to g/cm3
Convert 1 g/cm3 to kg/m3
Kinetic energy is defined as ½mv2 and has a unit of joule (J) where m is mass in kg and v is speed in m/s. A large object has a mass about 1600 Gg and move with a speed of 0.5 km/hr. Find the kinetic energy of this object in J?
Objectives of Physics
To find the limited number of fundamental laws that govern natural phenomenaTo use these laws to develop theories that can predict the results of future experimentsExpress the laws in the language of mathematics
Mathematics provides the bridge between theory and experiment.
Introduction
CH1
Math Review
Algebra : solve basic equation, exponent number, logarithmic
number, linear equation, solving simultaneous
equation, etc.
Scientific Notations.
Trigonometry and Geometry
Vector
Calculus : Derivative
and Integration
Scientific Notation
For very large and very small number, it becomes cumbersome to read, write and memorize.
How would you describe these numbers?70,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000.00and
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000001
Scientific Notation
For very large and very small number, it becomes cumbersome to read, write and memorize. We avoid this problem by using a method dealing with powers of 10.
Number of zeros
Scientific Notation
Numbers expressed as some power of 10 multiplied by another number between 1 and 10 are said to be in scientific notation.
Calculus
There are two components to calculus.
One is the measure the rate of change at any given point on a curve. This rate of change is called the derivative.
The second part is used to measure the exact area under a curve. This is called the integral.
The derivative and the integral are
inverse functions of each other.
http://www.wtv-zone.com/Angelaruth49/Calculus.html
Calculus: Derivative To measure the rate of change of
x is to calculate the slope
slope = y2-y1 = y
x2-x1 x
When x → 0,it become a measurement of the rate of change at any given point on a curve, or “the derivative”.
Calculus: Derivative
Calculus: Derivative
Exercise
Find the derivative of y(x), where 1. y(x) = x3 + 5x/(x+1)2. y(x) = Asin(5x) + Bcos(wx) where A, B and w are constant.3. y(x) = x2cos(x2)
4. Find the derivative of y = f(x) = x2 + 3x with respect to x = x0. Use this to find the value of the derivative at (a) x0 = 2 and (b) x0 = -4.
5. (a) y = x2 + 3x + 3; find dy/dx; (b) y = (x-5)2; find dy/dx and d2y/dx2.
6. (a) y = (3-2x)/(3+2x) , find dy/dx; (b) y = 1/x, find dy/dx and d2y/dx2.
Calculus : Integration Integration is the measure of the area under
a curve and an inverse of the derivative.
Integration from x1 to x2 is equal to the area under a curve
≈ x ½ (y2+y1)
When x → 0,it become an area under a curve at any given point on a curve, or “the integration”.
Calculus : Integration
Calculus: Integration
In physics, the students in fact need to know how to do calculus.
However, in this course, we try to minimize the use of calculus or only require “simple” calculus.
VectorsVector quantities
Physical quantities that have both numerical and directional properties
Mathematical operations of vectors in this chapter Addition Subtraction
Coordinate Systems used to describe the position of a point in space
Cartesian coordinate system Polar coordinate system
CH3
Cartesian Coordinate System
Also called rectangular coordinate systemx- and y- axes intersect at the originPoints are labeled (x,y)
Polar Coordinate SystemOrigin and reference line are notedPoint is distance r from the origin in the direction of angle , ccw from reference line
The reference line is often the x-axis.Points are labeled (r,)
CH3
Polar to Cartesian Coordinates &Cartesian to Polar CoordinatesBased on forming a right triangle from r and x = r cos y = r sin
If the Cartesian coordinates are known:
2 2
tany
x
r x y
Section 3.1CH3
Example 3.1The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point.
Solution: From Equation 3.4,
and from Equation 3.3,
2 2
2 2( 3.50 m) ( 2.50 m)
4.30 m
r x y
2.50 mtan 0.714
3.50 m216 (signs give quadrant)
y
x
Section 3.1CH3
Convert r and to x and y?
Vectors and ScalarsA scalar quantity is completely specified by a single value with an appropriate unit and has no direction.
Many are always positive Some may be positive or negative Rules for ordinary arithmetic are used to
manipulate scalar quantities.A vector quantity is completely described by a number and appropriate units (called magnitude) plus a direction.
Section 3.2CH3
Vector ExampleA particle travels from A to B along the path shown by the broken line.
This is the distance traveled and is a scalar.
The displacement is the solid line from A to B
The displacement is independent of the path taken between the two points.
Displacement is a vector.
Section 3.2
Force is a good example of vector quantity.
CH3
Vector NotationText uses bold with arrow to denote a vector: Also used for printing is simple bold print: AWhen dealing with just the magnitude of a vector in print, an italic letter will be used: A or | |
The magnitude of the vector has physical units. The magnitude of a vector is always a positive
number.When handwritten, use an arrow:
A
A
A
CH3
Magnitudes are always positive!
A vector can never equal a scalar. Never write
A = A .
Remember, vectors have both magnitude and direction. You can specify a vector by:
magnitude and direction (5 meters, northeast)
magnitude and angle it makes with some axis (5 meters, 45 counterclockwise from +x axis)
components with respect to axes.
If a problem requires a vector as an answer, your answer must provide information about both a magnitude and a direction.
Vector Notation
Equality of Two Vectors
Two vectors are equal if they have the same magnitude and the same direction. if A = B and they point along parallel linesAll of the vectors shown are equal.Allows a vector to be moved to a position parallel to itself
A B
Section 3.3
CH3
Adding VectorsVector addition is very different from adding scalar quantities.When adding vectors, their directions must be taken into account.Units must be the same Graphical Methods
Use scale drawingsAlgebraic Methods
More convenient
Section 3.3
CH3
Adding Vectors Graphically
Choose a scale. Draw the first vector, , with the appropriate length and in the direction specified, with respect to a coordinate system.Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for .
A
A
A
Section 3.3
CH3
Tip (or Head) to Tail Method for Adding Two VectorsPlace the tail of the second vector at the tip of the first vector. The resultant is the vector from the beginning tail to the ending tip.
A
B
A B
A+B
A
“Slide” vector B so that its tail touches A’s tip.
B
The resultant is drawn from the origin of the first vector to the end of the last vector.Measure the length of the resultant and its angle.
Use the scale factor to convert length to actual magnitude. Ref?
Parallelogram Method for Adding Two Vectors
The tail of the second vector is placed at the tail of the first vector. The two vectors define a parallelogram. The resultant is the vector along the diagonal of the parallelogram.
A
B A+B
A
“Slide” vector B so that its tail touches A’s tail.
A
B
Complete the parallel-ogram. The resultant is the diagonal.
B
The resultant is drawn from the origin of the first vector to the end of the last vector.Measure the length of the resultant and its angle.
Use the scale factor to convert length to actual magnitude. Ref?
Both the tip to tail and parallelogram method produce the same resultant.
A B
A+B A+B
A
B
The magnitude of the sum is always less than or equal to the sum of the magnitudes of the vectors being added; this may provide clues to incorrectly worked problems.
Ref?
What do you think of this:
A
B
1
A+B
A
B
3
A
B
2
BAD! WRONG! DO NOT TRY THIS AT HOME! (or in class, either)
You never saw that done here! Ref?
Adding Vectors Graphically, finalWhen you have many vectors, just keep repeating the process until all are included.The resultant is still drawn from the tail of the first vector to the tip of the last vector.
Section 3.3
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Adding Vectors, Rules
When two vectors are added, the sum is independent of the order of the addition.
This is the Commutative Law of Addition.
A B B A
Section 3.3
CH3
Adding Vectors, Rules cont.When adding three or more vectors, their sum is independent
of the way in which the individual vectors are grouped. This is called the Associative Property of Addition.
A B C A B C
Section 3.3
CH3
Adding Vectors, Rules final
When adding vectors, all of the vectors must have the same units.All of the vectors must be of the same type of quantity.
For example, you cannot add a displacement to a velocity.
Section 3.3
CH3
Negative of a Vector
The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero.
Represented as
The negative of the vector will have the same magnitude, but point in the opposite direction.
A
0 A A
Section 3.3
CH3
AA
Subtracting Vectors
Special case of vector addition:If , then useContinue with standard vector addition procedure.
A B
A B
Section 3.3
CH3
Subtracting Vectors, Method 2Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector.
As shown, the resultant vector points from the tip of the second to the tip of the first.
A B C
Section 3.3
CH3
Vector Multiplication by a Scalar
If a is a scalar then is a vector parallel to and a times the length of .
BC = 2 B
C = 0.5 B
C = -2 B
C = aB
B
B
can be longer than (if a>1) or shorter than (if a<1). If a is negative, then is in the opposite direction to .
C
B
B
B
C
Ref?
Multiplying or Dividing a Vector by a ScalarThe result of the multiplication or division of a vector by a scalar is a vector.The magnitude of the vector is multiplied or divided by the scalar.If the scalar is positive, the direction of the result is the same as of the original vector.If the scalar is negative, the direction of the result is opposite that of the original vector.
Section 3.3
CH3
Component Method of Adding VectorsGraphical addition is not recommended when:
High accuracy is required If you have a three-dimensional problem
Component method is an alternative method It uses projections of vectors along coordinate
axes
Section 3.4
CH3
Components of a VectorA component is a projection of a vector along an axis.
Any vector can be completely described by its components.
It is useful to use rectangular components.
These are the projections of the vector along the x- and y-axes.
Section 3.4
CH3
are the component vectors of .
They are vectors and follow all the rules for vectors.
Ax and Ay are scalars, and will be
referred to as the components of .
andx yA A
A
A
Components of a Vector
Assume you are given a vector It can be expressed in terms of two other vectors, and
These three vectors form a right triangle.
x y A A A
A
xA
yA
Section 3.4
The y-component is moved to the end of the x-component.This is due to the fact that any vector can be moved parallel to itself without being affected.
This completes the triangle.
CH3
Components of a VectorThe x-component of a vector is the projection along the x-axis.
The y-component of a vector is the projection along the y-axis.
This assumes the angle θ is measured with respect to the x-axis.
sinyA A
cosxA A
Section 3.4
The components are the legs of the right triangle whose hypotenuse is the length of A.
May still have to find θ with respect to the positive x-axis
2 2 1and tan yx y
x
AA A A
A
CH3
Components of a Vector, finalThe components can be positive or negative and will have the same units as the original vector.The signs of the components will depend on the angle.
Section 3.4
CH3
Unit VectorsA unit vector is a dimensionless vector with a magnitude of exactly 1.Unit vectors are used to specify a direction and have no other physical significance.
Section 3.4
The symbols
represent unit vectorsThey form a set of mutually perpendicular vectors in a right-handed coordinate system The magnitude of each unit vector is 1
kand,j,i
ˆ ˆ ˆ 1 i j k
CH3
Unit Vectors in Vector Notation
Ax is the same as Ax and Ay is the same as Ay etc.The complete vector can be expressed as:
ji
ˆ ˆx yA A A i j
Section 3.4
CH3
Position Vector, Example
A point lies in the xy plane and has Cartesian coordinates of (x, y).The point can be specified by the position vector.
This gives the components of the vector and its coordinates.
ˆ ˆˆ x y r i j
Section 3.4
CH3
Adding Vectors Using Unit VectorsUsing Then
So Rx = Ax + Bx and Ry = Ay + By
ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
x y x y
x x y y
x y
A A B B
A B A B
R R
R i j i j
R i j
R i j
R A B
2 2 1tan yx y
x
RR R R
R
Section 3.4CH3
y
x
Example 3.5 – Taking a HikeA hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.
Section 3.4CH3
Example 3.5 – Solution, Finalize
The resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east.The units of are km, which is reasonable for a displacement.From the graphical representation , estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of the resultant.
R
Section 3.4
Adding Vectors by using unit vectors
is a vector 66.0 units long at a 28 angle with respect to the positive x axis. is a vector 40.0 units long at a 56 angle with respect to the negative x axis. Calculate and give the resultant in terms of its (a) components and (b) magnitude and angle with the x axis.
B
A
A +B
= (Ax + Bx) î + (Ay + By) ĵ
= (58.27 + -22.37) î + (30.99 + 33.16) ĵ
= 35.9 î + 64.1 ĵ
C A B
Ax = A cos = 66.0 cos 28.0 = 58.27 Bx = - B cos = - 40.0 cos 56 = - 22.37Ay = A sin = 66.0 sin 28.0 = 30.99By = B sin = 40.0 sin 56 = 33.16
Cx = 35.9Cy = 64.1
Vector Multiplications
Dot product Cross product (will be later discussed when used)
Dot Product
Using in Work,Power, Electric flux, Electrical potential energy, etc. Properties of dot product
• 0 ≤ < 90 : A B > 0
• = 90 : A B = 0
• 90 < ≤ 180 : A B < 0
Dot Product
In Unit vector form
In special case:
Cross Product
Direction of C is perpendicular to AB plane
Using in Torque, angular momentum, magnetic force, magnetic field, etc.
Cross Product
Properties:
1.
2.
3.
4.
5.
Cross Product
In Unit vector form:
Vector calculus
Derivative of r w.r.t time is
I am skipping the rest of these slides, but you should review by yourself to make sure you can do the math!.
Math is important in learning Physics!
Algebra: Basic Equation
An equation is a statement of equality, i.e. both sides of the equation are equal.
Some equations contain unknown variables, such as x, either on the left side or right side or both. We must solve for its value which still make the statement correct.
Example 2.1.1: Solve 5x – 10 = 20.
Example 2.1.2: Find a from 5/(a+2) = 3/(a-2).
In all cases, whatever operation is performed on the left side of the equality must also be performed on the right side.
Algebra: Basic Equation
Algebra: Basic Equation
Algebra: Powers or Exponent number
Algebra: Powers or Exponent number
Algebra: Factoring
x2 + 6x + 9 = (x + 3)2
x2 – 6x + 9 = (x – 3)2
x2 – 16 = (x + 4)(x – 4)
Algebra: Quadratic Equation
Algebra: Quadratic Equation
Algebra: Quadratic Equation
Algebra: Linear Equation
Algebra: Linear Equation
If m > 0, the straight line has a positive slope.If m < 0, the straight line has a negative slope.
Algebra: Linear Equation
Algebra: Solving simultaneous equations
If a problem has two unknowns, a unique solution is possible only if we have two equations. In general, if a problem has n unknowns, its solution r
equires n equations.
In order to solve two simultaneous equations involving two unknowns, x and y, we solve one of the equations for x in terms of y and substitute thi
s expression into the other equation.
This equation has two unknown, x and y. Such an equation does not have a unique solution.
For example, x = 0 and y = 3 is a solution for this equation.
x = 5 and y = 0 is also a solution.
x= 2 and y = 9/5 is also another solution.
Algebra: Solving simultaneous equations
Algebra: Solving simultaneous equations
Algebra: Solving simultaneous equations
Geometry
Geometry
Geometry
Geometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
