Mechanics 105 Standards Dimensional analysis Unit conversions Order of magnitude calculations/estimation Significant figures Coordinates Vectors and scalars Problem solving Introduction and vectors (chapter one)

Mechanics 105 SI, esu, British Length SI: meter (m) esu: centimeters (cm) British: foot (ft) Mass SI: kilogram (kg) esu: gram (g) British: slug Time SI: second (s) British: second (s) esu: second (s) Standards – fundamental units of length, mass and time

Mechanics 105 Derived units All other units can be defined in terms of fundamental ones (length, mass, time and charge) e.g. Speed: (length/time) Force: (masslength/time 2 ) Energy: (masslength 2 /time 2 )

Mechanics 105 Dimensional analysis Resolving units in terms of fundamental units (L, M, T) and treating them algebraically to check calculations e.g. work is a force acting on an object over some displacement The work-kinetic energy theorem says the work will result in a change in the kinetic energy of the object. Work=forceL=M L 2 /T 2 Kinetic energy = ½ mv 2 = M L 2 /T 2

Mechanics 105 Dimensional analysis continued Another (more useful) example: Why is the sky blue? Scattering from particles in the atmosphere. Scattered electric field is proportional to Incident electric field (E 0 ) 1/r (r is the distance from the particle to the observation point) Particle volume So: And since the scattered irradiance (power/area) is proportional to the square of the electric field so the proportionality constant has to go as L -4 to make I/I 0 dimensionless. The only other possible length that comes into the problem is the wavelength of the light,. Therefore, the scattered irradiance will be proportional to -4, in other words, blue light (small ) will scatter much stronger than red light (large ), giving the scattered light a bluish color.

Mechanics 105 Concept test – dimensional analysis Hooke’s law for a spring tells us that the force due to a spring is -k  x, where k is a constant and  x is the displacement from equilibrium. If we also know that force causes acceleration according to F=ma (mass times acceleration) what are the dimensions of the constant k? 1. M/L 2. L·M/T 2 3. M/T 2 4. It’s dimensionless

Mechanics 105 Use a conversion factor: a fraction equal to one, with the units to be converted between in the numerator and denominator e.g inch = 25.4 mm  1.00 = (1.00 inch/25.4 mm) How many inches is 57.0 mm? Units must cancel! Unit conversions

Mechanics 105 Estimating and order of magnitude calculations Order of magnitude: literally – to precision of a power of 10 Approximate value of some quantity Useful for checking answers Example: If this room were filled with beer, how much would it weigh? 1 st estimate room width, length, height Then estimate the density of beer from known quantity (e.g. water density  1 g/ml

Mechanics 105 Significant figures Level of precision of a number (precision – how many decimal places accuracy – how close a measurement is to the true value) Simplest to determine in scientific notation – it is the total number of digits in the coefficient) The output of a calculation can never be more precise than input Rough rules of thumb Multiplication and division: result has same SF as lowest SF of inputs Addition and subtraction: result has SF according to smallest decimal places of terms

Mechanics 105 Significant figures continued Examples: x 5.7 = 28, or 27.9 ( ) = , or X 7 = 30 ! Best to work problem to end, then truncate to proper # of significant figures (avoid round off errors in intermediate steps).

Mechanics 105 Coordinate systems Cartesian (x,y) linear motion Polar (r,  ) angular motion, circular symmetry y x r 

Mechanics 105 Vectors and scalars Scalars: magnitude only (mass, time, length, volume, speed) Vectors: magnitude and direction (velocity, force, displacement, momentum) Vector math – resolution into (orthogonal) components = AyAy AxAx AyAy AxAx

Mechanics 105 Vector decomposition Most of the time (especially in mechanics) the two components are independent, i.e., you can separate a vector equation into two or three scalar equations

Mechanics 105 Vector decomposition Another common decomposition that we’ll use extensively in discussion angular motion uses the radial and tangential directions. e.g. acceleration along a curved trajectory atat arar

Mechanics 105 Vector Vector addition: graphically or algebraically (note that the origin of the vector doesn’t matter – this holds only for point objects) A B A B C

Mechanics 105 Vector subtraction Same as adding negative vector A B A -B C C=A-B

Mechanics 105 Vector multiplication Mulitplication by a scalar each component multiplied by scalar Inner, scalar or dot product – Chapter 6 – work Outer, vector or cross product – Chapter 10 – torque

Mechanics 105 Vectors – concept tests

Mechanics 105 Models and problem solving Model building – simplification of key elements of problem e.g. particle model for kinematics (real objects are not particles, but motion can be described as that of effective particle) Pictorial representation Graphical representation Mathematical representation