Transcript
Introduction and Mathematical Concepts
Physics is the study of energy and its interactions with matter.
The basic SI units(metric units) are:
Length - meter - based on the speed of light
Mass - Kilogram - mass of a particular platinum alloy cylinder
Time - Second - based on the frequency of electromagnetic waves emitted by Cesium 133
Temperature - Kelvin - same size as a Celsius degree
Electric current - Ampere - one coulomb of charge per second
Luminous Intensity - Candela - produces 1/683 watts per steradian.
Mole - unit of quantity - 6.02 X 1023 particles, hot dogs, etc.
The base units are not normally expressed in terms of other units.
Derived units are composed of base units.
Example
m/s2 or 1 Newton = 1 Kgm/s2
Metric prefixes mega(106) through micro(10-6) will be frequently used.
To convert from one unit to another, replace the original unit with its equivalent amount of the new unit.
Example
3 Kg = 3(1000g) = 3000 g
Dimensional analysis is the process of dividing out, combining and expanding units to check the validity of equations and substitutions.
Example
Let X (in meters) = ½V(m/s)t2(s)
m = (m/s)s2
m = ms indicates the equation is not correct since the resulting units must be the same on both sides of the equation for the equation to be valid.
Trigonometry - use of solutions of triangles to solve problems.
The trig functions and relationships you need to know are:
For a right triangle
sin? = opposite/hypotenuse
cos? = adjacent/hypotenuse
tan? = opposite/adjacent
h2 = ha2 + ho2
? = tan-1(opposite/adjacent)
For any other triangle that is not a right triangle:
Law of cosines
c2 = a2 + b2 - 2abcosC
Law of sines
(sinA)/a = (sinB)/b = (sinC)/c
In both instances, the capital letters represent the angles of a triangle and the lower case letters represent the sides opposite the corresponding angles.
A scalar quantity is one which has magnitude but no direction. An example is mass.
A vector quantity has both magnitude and direction. An example is weight.
In drawings, vectors are represented by arrows. The direction of the arrow is the direction of the vector and the length of the arrow represents the magnitude of the vector.
Vector Addition and Subtraction
When two vectors are added, both the size and direction of each vector must be taken into account when finding the resultant.
If they are in the same direction, add them numerically and assign their direction to the resultant.
If they are in opposite directions, subtract the smaller from the larger and assign the direction of the larger to the resultant.
If, when placed head to tail, they form an angle other than 0 or 180 degrees, draw a diagram and use trig functions to solve the resulting triangle for the magnitude of the resultant and the direction.
When vectors form a right angle, the Pythagorean theorem can be used to find the magnitude of the resultant. Then use the inverse tangent to find the direction.
When vectors do not form a right angle, the law of cosines is used to find the magnitude of the resultant and the law of sines is used to find the direction.
Vector subtraction is done the same way except the vector being subtracted is drawn in the direction opposite to its original direction.
In diagram (a), vector A added to vector B gives the resultant vector C.
In diagram (b), vector B is subtracted from vector C to give the resultant vector A.
Notice that the same triangle would be solved in both cases.
Vector Components
When someone is working with vectors in 2 dimensions, vectors can be resolved into two components at right angles to each other.
If the vector is in standard position(tail at the origin), the two components are along the X axis and the Y axis.
The two components of vector A are Ax and Ay. They are found using the cosine function for Ax and the sine function for Ay.
If Ay is moved so that it forms a right triangle with Ax and A, any of the right triangle trig functions may be used to solve the triangle.
The vector A may then be expressed as a magnitude and direction angle or using the X and Y components.
When component notation is used, unit vectors are included to indicate the axis along which each numerical value points.
If the vector A has components Ax and Ay, it can be written like this:
A = Axî+ Ay?
î and ? represent the unit vectors along the X axis and Y axis respectively.
To find the magnitude of a vector component along X, multiply the magnitude of the vector by the cosine of the angle made by the vector and the nearest X axis.
Ax = Acos?
If A is located in the second or third quadrant, Ax is negative.
To find the magnitude of a vector component along Y, multiply the magnitude of the vector by the sine of the angle made by the vector and the nearest X axis.
Ay = Asin?
If A is located in the third or fourth quadrant, Ay is negative.
Addition of vectors by means of X and Y components
1. Find the X components and Y components of each of the vectors.
2. Find the algebraic sum of all the X components. Do the same for the Y components.
3. Use the resultant X component and resultant Y component in the Pythagorean Theorem formula to find the magnitude of the resultant.
4. Use the tan-1 function with the resultant X and Y components to find the direction angle.
Example
In diagram (a), two vectors A and B are added to give their resultant, vector C. The magnitude of A is 145m and the magnitude of B is 105m.
First find the X components and add them.
Ax = 145mcos70° = 49.6m
Bx = 105mcos35° = 86.0m
Cx = 49.6m + 86.0m = 135.6m
Then find the Y components and add them.
Ay = 145msin70° = 136m
By = 105msin(-35°) = -60.2m
Cy = 136m + (-60.2m) = 76.2m
Find the magnitude of C
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C = ?(135.6)2 + (75.8)2 = 155m
Find the direction angle.
? = tan-1(75.8m/135.6m) = 29.2°
The vector C is then expressed as 155m at 29.2° N of E.
