Unformatted text preview: Derivatives
Polynomials: Trig functions: Exponential, log: Product rule: Chain rule: dn x = nx n−1 dx
d d sin ( ax ) = a cos ( ax ) cos (ax ) = − a sin (ax ) dx dx Physics for Scientists & Engineers 1
Spring Semester 2011
Lecture 1  Math Primer Appendix A, Chapter 1 d ax e = ae ax dx d 1 ln ( ax ) = dx x dx a = a x ln a dx d df ( x) dg ( x) ( f ( x) g ( x ) ) = g ( x) + f ( x) dx dx dx y(u ( x)) ⇒ dy dy du = dx du dx January 12, 2011 Physics for Scientists&Engineers 1 1 January 12, 2011 Physics for Scientists&Engineers 1 2 Integrals
Polynomials: Geometry ∫ x dx = n + 1 x
n 1 n +1 + c for n ≠ 1 ∫x −1 dx = ln x + c A = area, V = volume, C = circumference ∫a 2 1 1 x dx = tan −1 + c + x2 a a 1 ∫ 1
a2 − x2 x dx = sin −1 + c a Trig functions: ∫ sin ( ax ) dx = − a cos ( ax ) + c 1 ∫ cos ( ax ) dx = a sin ( ax ) + c
ax Exponential: ∫e dx = 1 ax e +c a January 12, 2011 Physics for Scientists&Engineers 1 3 January 12, 2011 Physics for Scientists&Engineers 1 4 Trigonometry
(Right) triangles appear in many places in introductory physics problems
• It pays to remind ourselves of some basic trigonometry Inverse Trigonometric Functions
We can invert the trigonometric functions, which is useful in many physical situations
a a sin −1 ≡ arcsin = α c c cos−1
b b ≡ arccos = α c c sin α = a opposite side = c hypotenuse b adjacent side cos α = = c hypotenuse
tan α =
January 12, 2011 sin α a = cos α b cot α =
Physics for Scientists&Engineers 1 cos α 1 b = = sin α tan α a
5 a a tan −1 ≡ arctan = α b b b b cot 1 ≡ arccot = α a a
6 January 12, 2011 Physics for Scientists&Engineers 1 1 Triangles
Right triangles:
• Pythagoras Scientific Notation
Physical quantities consist of a number that specifies its magnitude AND its unit
• Example: this lecture lasts 50 minutes (number) (unit) a2 + b2 = c2
• Same, using trig functions sin 2 α + cos 2 α = 1
General triangles
• Law of cosines For very large or very small numbers, we use scientific notation c 2 = a 2 + b 2 − 2 ab cos γ number = mantissa ⋅ 10 exponent • Example: 3.2·1012 (or 3.2x1012)
• Product easy: (4.8x1017)x(7.21x107)=34.6x1010=3.46x109 • You can enter number in scientific notation into the LONCAPA homework system as 3.2e12 or 3.2*10^12 • In general, angle sum is 1800 α +β +γ =π
January 12, 2011 Physics for Scientists&Engineers 1 7 January 12, 2011 Physics for Scientists&Engineers 1 8 Significant Figures  Rules
The number 1.62 has 3 significant digits The number 1.6 has 2 significant digits If you give a number as an integer, then you specify it with infinite precision Leading zeros do not count for our significant digits
• 0.000162 (3 sd) Units  The Metric System
International system of units (SI)
Unit Meter kilogram Second Ampere Kelvin Mole candela Abbreviation m kg s A K mol cd Base unit for length mass time current temperature amount of a substance luminous intensity Trailing zeros do count
• 1.62000 (6 sd) Numbers in scientific notation have as many significant digits as their mantissa
• How big the exponent is has no influence You can never have more significant figures than you start with in any of the factors of a multiplication or division You can only add or subtract when there are significant figures for that place in every number
• Example: 1.23 + 3.4461 = 4.68, and not 4.6761 MKSA system Based on powers of 10 of base units All other units derive from these 7 units (Area: m2)
January 12, 2011 Physics for Scientists&Engineers 1 10 January 12, 2011 Physics for Scientists&Engineers 1 9 Definition of Base Units
1 kilogram of mass is defined as the mass of the international prototype of the kilogram kept in Paris 1 second is the time interval in which 9,192,631,770 oscillations of the wave that corresponds to the transition between the two hyperfine states of the ground state of the cesium133 atom take place 1 meter is the distance that light in a vacuum travels in a fraction of 1/299,792,458 of a second Powers of Ten Prefixes
10 24 yotta Y 10 21 zetta Z E 1018 exa 15 peta P 10 1012 tera T 10 9 giga G 10 6 mega M kilo k 10 3 2 hecto h 10 101 deka da
11 January 12, 2011 10 −24 yocto 10 −21 zepto − 18 atto 10 10 −15 femto 10 −12 pico 10 − 9 nano 10 − 6 micro 10 − 3 milli 10 − 2 centi 10 − 1 deci y z a f p Case n Sensitive! µ m c d
12 January 12, 2011 Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 2 Metrology
Research on precision measurements Atomic clocks accurate to 1015 (= 1 s in 30 million years) Precision needed for global positioning system (GPS) Main US research institute: National Institute for Standards and Technology (NIST) Length Scales Cesium Fountain Atomic Clock at NIST • Attractive career option for physicists and engineers • URL: http://www.nist.gov/public_affairs/labs2.htm January 12, 2011 Physics for Scientists&Engineers 1 13 January 12, 2011 Physics for Scientists&Engineers 1 14 Mass Scales Vectors
Vectors are heavily used in physics Need to manipulate them without any difficulty Vector C
• Beginning and end point • Characterized by: Magnitude Direction Unit Remark: a quantity defined without a direction is a scalar January 12, 2011 Physics for Scientists&Engineers 1 15 January 12, 2011 Physics for Scientists&Engineers 1 16 Cartesian Coordinate System
Quantifies a direction in 3dimensional space
Third direction coming straight out of page RightRightHand Rule Higher dimensionalities are used in modern theories (but quite abstract and hard to represent on a twodimensional paper) Conventional assignment for righthanded coordinate system (more on 3D coordinate systems later in this semester)
January 12, 2011 Physics for Scientists&Engineers 1 17 January 12, 2011 Physics for Scientists&Engineers 1 18 3 Cartesian Coordinate System
In the Cartesian representation we define a displacement vector as the difference in the coordinates of the end point and the starting point. Only the difference matters We can shift the vector around as much as we like. Vector Addition  Graphical
C = A+ B
We learned: you can drag vectors around in space without changing their value
• • Length stays the same Direction stays the same In particular, you can drag vector B in such a way that its foot is at the tip of vector A Sum vector C then points from the foot of A to the tip of B You can do it the other way around C = A+ B = B + A
January 12, 2011 Physics for Scientists&Engineers 1 19 January 12, 2011 Physics for Scientists&Engineers 1 20 Vector Subtraction
For every vector A there is a vector − A, with the same length, pointing in the exact opposite direction A −A A + ( − A) = 0 Vector subtraction: y To obtain the vector D = B − A , we add the vector − A to B , following the procedure of vector addition. In Vector Subtraction Order Matters
Reverse the order and take A − B instead of B − A . What is the result? The resulting vector E = A− B is exactly the opposite vector to D = B − A Rules for vector addition and subtraction are just like for real numbers
21 January 12, 2011 y B −A x D = B−A
y AB −B −A A x
22 x D = B−A E = A− B January 12, 2011 Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 Unit Vectors
Vector representation in terms of unit vectors: Component Method for Vector Addition
ˆ x = (1,0, 0) ˆ y = (0,1,0) z = (0,0,1) ˆ Vector addition can also be accomplished by using Cartesian components and unit vectors. Component representation ˆ ˆ ˆ A = ax x + a y y + az z
ˆ ˆ ˆ B = bx x + by y + bz z Vector addition C = A+ B ˆ ˆ ˆ ˆ ˆ ˆ = [a x x + a y y + az z ] + [bx x + by y + bz z ] ˆ ˆ ˆ A = ax x + a y y + az z
2D case y
Projection of A on the y axis provides its component ay A ˆ ˆ A = ax x + a y y
ˆ ax x
ˆ x ˆ ay a y y ˆ y ˆ ˆ ˆ = (ax + bx ) x + ( a y + by ) y + (az + bz ) z Components of sum vector
ˆ ˆ ˆ C = cx x + c y y + c z z x
23 with c x = a x + bx cy = ay + by cz = az + bz ax
January 12, 2011 Physics for Scientists&Engineers 1 January 12, 2011 Physics for Scientists&Engineers 1 24 4 Addition of Two 2D Vectors Vector Subtraction
Procedure is exactly the same as vector addition
ˆ ˆ ˆ A = ax x + a y y + az z ˆ ˆ ˆ B = bx x + by y + bz z Difference vector: D = A− B ˆ ˆ ˆ ˆ ˆ ˆ = [ax x + a y y + az z ] − [bx x + by y + bz z ] ˆ ˆ ˆ = ( ax − bx ) x + (a y − by ) y + (az − bz ) z With components:
d x = ax − bx
ˆ ˆ A = Ax x + Ay y ˆ ˆ B = Bx x + By y ˆ ˆ ˆ ˆ C = C x x + C y y = A + B = ( Ax + Bx ) x + ( Ay + By ) y
January 12, 2011 Physics for Scientists&Engineers 1 25 ˆ ˆ ˆ D = dx x + dy y + dz z with d y = a y − by d z = az − bz An equation between vectors equals three scalar equations!
January 12, 2011 Physics for Scientists&Engineers 1 26 Multiplication of a Vector with a Scalar
Let imagine adding the same vector to itself three times
A+ A+ A Vector length and direction
Vector A in component representation (in 2D)
ˆ ˆ A = ax x + a y y The resulting vector is three times longer and points in the same direction as the original vectors For multiplication of a vector with a scalar, we obtain
E = sA = s ( Ax , Ay , Az ) = ( sAx , sAy , sAz ) Calculation of its norm (=length) from its components y ˆ ay y
ˆ y P
A Using Pythagoras in the right triangle OPQ The components are E x = sAx θQ
x A = ax 2 + a y 2
Also, the angle E y = sAy E z = sAz
January 12, 2011 Physics for Scientists&Engineers 1 27 O x ax ˆ xˆ θ between A and the x axis θ = sin −1 (a y / ax )
January 12, 2011 Physics for Scientists&Engineers 1 28 5 ...
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 Spring '08
 Wolf
 Physics, Trigonometry, Scientific Notation, Cartesian Coordinate System, René Descartes

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