Units and Vectors Lecture

Units and Vectors ● ● ● Unit: measure of a quantity that scientists around the world can refer to. ○ Must be accessible (easy to measure) and invariable (standard, the same everywhere, and every time). ○ Ex. 1 meter (m) is a unit of length. Any other length can be expressed in terms of 1 meter. A variable length, such as the length of a person’s nose, is not appropriate because it changes. SI system: international system of units. ○ AKA the metric system ○ 3 basic quantities are length (meter, m), time (second, s), and mass (kilogram, kg) ○ Many other units (eg. for speed, m/s; force kg*m/s2) can be derived from these ○ US uses other units, but this class focuses on SI system Prefixes ○ Correspond to powers of 10 ○ Each has a specific name ○ Each has a specific abbreviation ○ Scientific notation expresses big numbers using powers of 10 ■ 3, 560,000,000 m = 3.56 x 109 meters, or 3.56 gigameters ○ ● ■ Another good one to know is angstrom (1010) Basic Quantities and their Dimensions ○ Dimension denotes the physical nature of a quantity ○ Denoted with square brackets ○ Each dimension can have many actual units ○ ○ ● ● Dimensional Analysis checks the correctness of an equation or assists in deriving the equation if you forget a formula ■ Dimensions can be treated like algebraic quantities ■ Both sides of an equation must have the same equations ● Velocity (m/s) = displacement (m) / time (s) ■ Limitations: dimensional cannot give numerical factors (can’t tell you whether a formula needs coefficients/constants) Symbols ○ Symbol in equation might be different from symbol in dimension ○ Some quantities have quite standard symbols (ex. t=time) ○ Some quantities have different symbols per situation (ex. Length may be x, y, z, r, d, h, etc) Conversion of Units ○ When units aren’t consistent (ex. Feet and meters), convert to SI units ○ Chain-link conversion ■ ■ ■ ● ● ● ● ● The first equation (1 min/60 s = 60 s/1 min) is called conversion factor. Multiply the value you have (2 min) by the part of the conversion factor that has the same unity in the denominator (60 s/1 min) and multiply through/cancel out where necessary. The meter is now defined in respect to the second; it is the length of the path traveled by light in 1/299,792,458 of a second) Time ○ The second is the most accurate SI unit of measurement. ○ Atomic clocks give very precise time measurements ○ The standard second is defined by the time taken by 9,192,631,770 oscillations of light emitted by cesium-133 atom. Mass ○ A platinum-iridium cylinder has the standard mass of 1 kg. ○ 1 u is used for atomic mass measurements. ■ 1/12 of a carbon-12 atom’s mass ■ 1.66053886x10-27 kg Uncertainty ○ Due to sources of error (apparatus, experimenter, repeated measurements), exact measurements are uncertain ○ Significant figures help Significant Figures ○ Significant figure: a digit that is reliably known and is not a placeholder ○ All nonzero numbers and the digits between them are significant ○ Leading zeroes (zeroes before decimal) are placeholders, and do not count as sig figs ○ ○ Zeroes after decimal are significant because they indicate reliability of the measurement Some ambiguous examples: ■ 1500 m is ambiguous ● 1.5 x103 ● 1.50 x103 ● 1.500 x103 Vectors ● Coordinate Systems ○ Used to describe position of a point in space ○ Consists of ■ Origin: fixed reference point ■ Specific axes with scales and labels ■ Instructions on how to label a point relative to the origin and the axes ○ Cartesian ■ Rectangular coordinate system ■ X- and y- axes intersect at origin ■ Points are labeled (x,y) Coordinate System ■ ○ Polar Coordinate System ■ Origin and reference line are noted ■ Point is distance r from the origin in the direction of angle theta, counterclockwise from reference line ■ Points are labeled (r, θ(theta)) ● ■ Conversions ○ Polar to Cartesian coordinates ■ Based on forming a right triangle from r and θ ■ x= rcosθ ■ y= rsinθ ■ ○ Cartesian to Polar Coordinates ■ r is the hypotenuse and theta an angle ■ ■ Theta must be ccw from positive x axis for these equations to be valid ● ■ Adding Vectors Graphically ○ Continue drawing the vectors “tip-to-tail” ○ The resultant is drawn from the origin of ○ ● to the end of the last vector Measure the length of and its angle ■ Use the scale factor to convert length to magnitude ○ ○ Rules ■ Commutative/Associative Property of Addition: When 2 or more vectors are added, the sum is independent of the way they are grouped/added Negative of a Vector ○ Negative of a vector: when added tot the original vector, gives a resultant of zero ○ Represented as ○ ● ○ Has the same magnitude, but opposite direction Subtracting Vectors ○ If ● , then you can just use ○ Unit Vector ○ Unit Vector: dimensionless vector with magnitude of exactly 1. ○ Used to specify direction and have no other physical significance ○ These symbols represent unit vectors: ■ ○ They form a set of mutually perpendicular vectors in a right-handed coordinate system ● ○ Remember, Components of a Vector, Intro ○ Component: a projection of a vector along an axis ■ Any vector can be completely described by its components ■ Using the rectangular components, the projections of the vector along the x- and y- axis, is useful ■ ■ A= Axi+Ayj ○ Adding using unit vectors ■ Using ■ Then ■ And so Rx = Ax + Bx and Ry = Ay + By Motion in One Dimension: Part 1 ● ● ● Kinematics ○ Describes motion while ignoring the objects (and reasons) that caused the motion ○ For now, will consider motion in one dimension, along a straight line ○ Will use the particle model ■ Particle: point-like object, has mass but infinitesimal size Position ○ Position: location with respect to a chosen reference point ■ That point may be the origin of a coordinate system ○ Remember units and direction ● Position-Time Graph (x → t graph) ○ Shows the motion of the particle ○ Based on a data table, like such: ○ ○ ○ ● ● ● The smooth curve connects the data points, assuming what happened between them ■ Displacement: Delta(Δ) x is change in position. Has magnitude and direction. Vector xfinal-vector xinitial ■ Delta t is change in time. Alternative Representations ○ Pictorial, tabular, etc. ○ Goal is often a mathematical representation More on displacement ○ Displacement: Delta x. change in position during some time interval. Has magnitude and direction. ○ Displacement vector: Vector xfinal-vector xinitial ○ Si units are meters ○ Can be positive or negative ○ Different from distance: a scalar; length of path followed by particle ■ Always positive Average Velocity ○ Rate at which displacement occurs ○ ○ ○ ○ ■ X indicates motion along the x axis Dimensions are length / time [L/T] SI units are m/s Is also the slope of the line in the position/time graph Motion in One Dimension: Part II ● ● ● *remember arrows when writing vector quantities Instantaneous velocity: indicates what is happening at every point of time ○ The limit of the average velocity as the time interval approaches zero ○ Can be positive, negative, or zero Graph ○ =slope of the tangent line (green line) to x vs. t curve ○ ○ ● ● The blue secant lines, which represent average acceleration on those time intervals as they get closer and closer to the green line (dt gets smaller), approximate it ○ Slopes on the x vs t graph are the values on the v versus t graph. Instantaneous velocity, or just velocity, is constant when the x vs t line is straight. Particle Under Constant Velocity ○ When instantaneous velocity t any instant during a time interval = the average velocity during the time interval, or instantaneous velocity doesn’t change ○ ■ Or xf = xi + vx t ○ Constant velocity isn’t too common because of friction ○ ■ ■ ● slope= instantaneous and average velocity Y intercept is xinitial Acceleration ○ Average acceleration: rate of change of velocity ○ ○ ○ ○ ○ Measured in [L]/[T2] or m/s2 Positive and negative can indicate direction acceleration/instantaneous acceleration: limit of average acceleration as dt approaches 0. Slope of vt graph is acceleration Motion in 1D III ● ● ● ● Acceleration vs Velocity ○ When they are in the same direction, object is speeding up ○ When they are in different directions, object is slowing down Important Kinematic Equations ○ ○ Need to know how they are derived, and how they are applied ○ Can be used with any particle under uniform acceleration moving in one direction a=0 ○ When acceleration is 0, velocity is constant, and xf= xi + vxt Dt curve with constant acceleration ○ slope of the curve would be velocity ○ A curved line indicated that a changing velocity, or an acceleration, is present ● ● ● ● ● ● Examples ○ ○ a-e, b-d, c-f Free Fall ○ Freely falling object: any object moving freely under the influence of gravity alone. ○ Doesn’t depend upon initial motion of object ○ Language: ■ Dropped: released from rest, or vi=0 ■ Thrown downward/upward: vi is negative/positive ■ Acceleration is negative in free fall. Acceleration in the y direction (ay) can now be represented by -g, which is acceleration due to gravity (g) Acceleration in free fall ○ Acceleration is directed downward, no matter what ○ g= 9.80 m/s2 ■ This is the average at the Earth’s surface ■ All objects fall at the same rate when the force of friction (air resistance) is ignored Object is dropped ○ Initial velocity is 0, -g = -9.80 m/s2 ○ Use the kinematic equations, subbing in y for x since you are dealing with vertical direction Object thrown upward ○ Object will move up slower and slower, pause, then fall faster and faster ○ Acceleration is still always zero ○ Velocity in the first half and second half of motion is symmetrical. It is mirrored in the negative direction ○ Instantaneous velocity at maximum height is 0 Example ○ ○ ○ ■ ■ ■ ■ vi= 12 m/s a= -9.80 m/s2 vf= 0 m/s vf= vi + at 0= 12 + (-9.80)(t) t= 1.22 s ■ ■ ■ ■ ■ Let height of hand → y=0 yi= 0 t= 1.22 s vi= 12 m/s vf= 0 m/s yf= yi + ½ (vi+vf)t yf= 0 + ½ (12+0)(1.22) yf= 7.35 m/s ○ Motion in 2D, I ● ● ● Kinematic Equations from Calculus ○ Displacement + area under vt curve. Integral from ti to tf of v(t) Motion in 2 Dimensions ○ If motion is in more than one dimension, + or - signs is not sufficient ■ Using vectors is better. Must now use full vector notation Position and Displacement ○ Position is described by a position vector ○ Displacement is defined by change in position ○ ○ (resultant = components added together) ○ ○ ○ To subtract final from initial, draw ri in the opposite direction (head to tail) and add rf and -ri. Example: ■ ■ ● *when evaluating inverse trig functions, be aware that trig functions have the same value in 2 quadrants, and the calculator only gives answers from one of these 2 quadrants (normally 1st or 4th quadrant) ■ ■ ● ■ Average Acceleration ○ Change in velocity is found by vector subtraction ○ ● ● ○ Average acceleration is a vector quantity directed along ○ Producing Acceleration ○ Whenever magnitude and/or direction of velocity changes ■ For example, when running on a circular track, velocity changes constantly Kinematic Equations (for constant acceleration) ○ Motion in 2 dimansions can be modeled as 2 independent motions in each of the 2 perpendicular directions associated with the x and y axis ○ Any influence in y direction does not affect motion in the x direction ○ Position vector is the sum of initial position, displacement resulting from initial velocity, and displacement resulting from acceleration