So the problematic number in scientific notation is
\[\frac{1}{10000000000000000000}=10^{-19}\]
Thus the electron charge in scientific notation is
\[q=1.6 × 10^{-19}C\]
Test Yourself
Attempt Number
Physical Quantity Value
Write in Scientific Notation
1
Pressure inside ocean depth
\[98700000 pa\]
\[p=......\]
2
Charge of an iron sphere
\[0.0000007 C\]
\[q=......\]
3
Speed of light in vacuum
\[300000000\frac{m}{s}\]
\[𝜗=......\]
4
Atomic radius is approximately
\[0.0000000025 m\]
\[r=......\]
: Significant Figures
These are digits that can be measured using measuring devices and contain real and estimated digits
Example: The length of a pen measured by a ruler (using a device) is a significant figure
While the number of 10th grade students is not significant as it wasn't measured with a device
(13.85 cm) Pen length
(13.8 cm) is a real number that all students would read the same
(5) is an estimated number that varies from student to student
Solved example: This device is called a galvanometer used to measure current and voltage. Now it's being used to measure current. What is the current measurement?
(2.3) Reading
Note there are two significant figures
: Rules of Significant Figures
Rule 1: All non-zero digits are significant
Example: 3.46 has three significant figures
Rule 2: Zeros to the left of the number are not significant
Example: 0.00034 has two significant figures (3, 4)
Rule 3: Zeros to the right of the number are not significant
Example: 5800 has two significant figures (5, 8)
Rule 4: Zeros between numbers are significant
Example: 5003 has four significant figures (5,0,0,3)
Rule 5: Zeros after the decimal are significant
Example: 6.00 has three significant figures (6,0,0)
We use prefixes to measure lengths in multiples and fractions
Common measurement units used include inches, feet, and miles
\[1 in=2.54 cm, 1 cm = 0.393 in\]
\[1 foot=0.3048 m, 1 m=3.28 foot\]
\[1 mile=1609 m, 1 m=6.215 ×10^{-4}mile\]
We use the astronomical unit to measure long distances
\[1 AU=1.49598 ×10^{11} m\]
We use the light-year unit to measure distances outside our solar system
\[1 Light-Year= 1 ×365.25 ×24 ×60×60 × 3 ×10^8=9.46 ×10^{15} m\]
Mass Measurements
Mass: The amount of matter in an object
The basic unit used to measure mass is the kilogram, used to measure very small masses like the mass of an electron
\[m_e=9.11 ×10^{-31} Kg\]
We use the kilogram unit to measure massive objects like the mass of the Earth
\[m_{earth}=6 ×10^{24} Kg\]
Time Measurements
Time: The duration between two events
We use the second as the international unit of measurement
For long time periods we use minutes, hours, days, years, and centuries
\[1min=1×60 s=60 s\]
\[1 hour=1×60×60=3600 s\]
\[1 Day=24×60×60=86.4×10^3 s\]
\[1 year= 1 ×365.25 ×24 ×60 ×60 =31.557 ×10^6 s\]
\[1 century=1 ×100×365.25 ×24 ×60 ×60 =31.557 ×10^8 s\]
5
Ahmed's height is \[(4foot ,20 in)\]. What is Ahmed's height in meters?
Vectors
Physical quantities are divided into two types:
First type: Scalar quantities
Determined by knowing their magnitude only
Examples: Distance - Mass - Volume - Time
Second type: Vector quantities
Determined by knowing their magnitude and direction
Examples: Displacement - Velocity - Acceleration - Force
Cartesian Coordinates
To determine the position of a point on a two-dimensional plane, we use two numbers called Cartesian coordinates
The first number is the distance along the horizontal axis
The second number is the distance along the vertical axis
\[(12.5 ,5 )\]
To determine the position of a point in three-dimensional space, we use three numbers called Cartesian coordinates
\[(P_X,P_Y,P_Z)\]
The first number is the distance along the horizontal axis \[X\]
The second number is the distance along the vertical axis \[Y\]
The third number is the distance along the axis perpendicular to the first two axes \[Z\]
Vector Components
The process of projecting a vector onto the horizontal axis and vertical axis, converting one vector into two components equivalent to this vector
\[𝐴_𝑋= A . Cos 𝜃\]
\[𝐴_Y= A . sin 𝜃\]
Resolution is the opposite process of vector resultant
When resolving, the sign of each component must be determined according to the axis it is projected on
Vector Components Calculator
Conclusions
The practical importance of vector analysis appears in:
Designing mechanical systems
Motion analysis in video games
GPS positioning systems
8
Vector \[A\] has a starting point with coordinates \[-5,-3\] and an ending point with coordinates \[5,1\]. Determine the coordinates of vector \[A\]
\[\overrightarrow A = +8 \widehat X ,+ 4\widehat Y -C\]
\[\overrightarrow A = +6 \widehat X ,+ 5\widehat Y -A\]
\[\overrightarrow A = +12 \widehat X ,+ 6\widehat Y -D\]
\[\overrightarrow A = +10 \widehat X ,+ 4\widehat Y -B\]
Vector Addition and Subtraction
Adding two or more vectors graphically:
We move one vector until its tail coincides with the head of the first vector (while maintaining magnitude and direction)
Then we connect from the tail of the first vector to the head of the second vector to get the required resultant
This method is used to find the resultant of more than two vectors
It doesn't matter which one we move first, the result is the same
Resultant of subtracting two vectors \[\vec A-\vec B\]
Using the triangle method:
First we reverse the direction of the subtracted vector to get vector (-B)
Then we perform the addition \[A+(-B)\] as previously explained
\[C=A+-B\]
In this simulation, you can find the sum and difference of two vectors graphically
(Degrees) The arrow indicator to change the angle in degrees
(Tail to tail) The top icon on the left shows vectors tail to tail
(Head to tail) The bottom icon on the left shows vectors head to tail
(Show resultant) The middle top icon on the left shows the sum resultant
(Show equilibriant) The middle top icon on the right shows the difference resultant
(Hide resultant) The middle bottom icon on the left hides the sum resultant
(HIDE equilibriant) The middle bottom icon on the right hides the difference resultant
Vector Addition and Subtraction Using Components
\[\vec A (𝐴_X , 𝐴_𝑌 , 𝐴_Z)\]
\[\vec B (B_X , B_𝑌 , B_Z)\]
When asked to find \[\vec C =\vec A +\vec B\], the solution method is:
\[C_X=A_X+B_X\]
\[C_Y=A_Y+B_Y\]
\[C_Z=A_Z+B_Z\]
When asked to find \[\vec C =\vec A -\vec B\], the solution method is:
\[C_X=A_X-B_X\]
\[C_Y=A_Y-B_Y\]
\[C_Z=A_Z-B_Z\]
Multiplying a Vector by a Scalar
When a scalar is multiplied by a vector, the result is a vector quantity
There are two cases:
First case:
If the scalar is positive, the product is a vector in the same direction as the original vector
Example: \[\vec A ( 2x , 4y ,-2z) \], find \[ 2\vec A\]
\[2\vec A=( 4x , 8y ,-4z) \] with the same direction as the original vector
Second case:
If the scalar is negative, the product is a vector in the opposite direction to the original vector
Example: \[\vec A ( 2x , 4y ,-2z) \], find \[ -2\vec A\]
\[-2\vec A=( -4x , -8y ,4z) \] with direction opposite to the original vector
Unit Vectors
These are vectors with magnitude 1 located on the axes
Vector Magnitude and Direction
If the components of a vector are known, we can determine its magnitude and direction
Example: Vector (A=60) making an angle of 20 degrees north of east
The vector has two components:
\[A_X=A .cos 𝜃= 60 COS (20)=56.38\]
\[A_y=A .sin 𝜃= 60 sin (20)=20.52\]
The vector is written as \[A(56.38 x,20.52y)\]
\[A= \sqrt {56.38^2+20.52^2}=60\]
The direction \[𝜃 =tan^{-1}\frac{20.52}{56.38}=20^0\]
Dot Product of Vectors
Work is a scalar quantity resulting from the dot product of the force vector and the displacement vector
\[W=|\vec F |.|\vec d |cos 𝜃\]
The dot product is defined as the magnitude of the first vector times the magnitude of the second vector times the cosine of the angle between them
\[\vec A.\vec B=|\vec A |.|\vec B |cos 𝜃\]
The dot product is denoted by a dot symbolizing the scalar product
Dot Product in Three Dimensions
1. Definition of Dot Product:
The dot product is an algebraic operation between two vectors that produces a scalar quantity.
It is calculated in two ways:
A. Geometric method:
A · B = |A| |B| cosθ
where θ is the angle between the vectors,
|A| is the magnitude of A, |B| is the magnitude of B
B. Algebraic method (using vector components):
If: A = Axî + Ayĵ + Azk̂ B = Bxî + Byĵ + Bzk̂
A · B = AxBx + AyBy + AzBz
2. Properties of Dot Product:
Commutative: A · B = B · A
Distributive: A · (B + C) = A · B + A · C
If vectors are perpendicular: A · B = 0
3. Practical Example:
Let: A = 3î + 4ĵ + 5k̂ B = î - 2ĵ + 2k̂
(3×1) + (4×-2) + (5×2) = 3 - 8 + 10 = 5
9
\[\vec A=2x+3y+4z\] \[\vec B=x-2y+3z\] Find the angle between the two vectors
Magnetic force is a vector quantity resulting from the cross product of the charge magnitude, velocity vector, and magnetic field vector
\[\vec F_B =- q.|\vec v |.|\vec B |sin 𝜃\]
The cross product is defined as the magnitude of the first vector times the magnitude of the second vector times the sine of the angle between them
The result is a vector quantity determined by the right-hand rule
\[\vec A×\vec B=|\vec A |.|\vec B |sin 𝜃\]
Cross Product in Three Dimensions
Definition of Cross Product:
The cross product between two vectors a and b in three-dimensional space gives a new vector:
a × b = |a||b| sin(θ) n̂
where:
θ: Angle between the vectors n̂: Unit vector perpendicular to both vectors
Mathematical Representation Using Unit Vectors:
If: a = ax𝐢 + ay𝐣 + az𝐤 b = bx𝐢 + by𝐣 + bz𝐤
a × b =
𝐢
𝐣
𝐤
ax
ay
az
bx
by
bz
Calculation is as follows:
= 𝐢(aybz - azby) - 𝐣(axbz - azbx) + 𝐤(axby - aybx)
Physics
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