Introduction and vectors (chapter one)

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

2. Standards – fundamental units of length, mass and time • 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)

3. Derived units • All other units can be defined in terms of fundamental ones (length, mass, time and charge) • e.g. • Speed: (length/time) • Force: (mass•length/time2) • Energy: (mass•length2/time2)

4. 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=force•L=M L2/T2 • Kinetic energy = ½ mv2 = M L2/T2 

5. 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 (E0) • 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/I0 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.

6. 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/T2 • 3. M/T2 • 4. It’s dimensionless

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

8. 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? 1st estimate room width, length, height • Then estimate the density of beer from known quantity (e.g. water density  1 g/ml

9. 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

10. Significant figures continued • Examples: • 4.892 x 5.7 = 28, or 27.9 (27.8844) • 5.0043 + 10.547 = 15.551, or 15.5513 • 4 X 7 = 30 ! • Best to work problem to end, then truncate to proper # of significant figures (avoid round off errors in intermediate steps).

11. y r  x Coordinate systems • Cartesian (x,y) • linear motion • Polar (r, ) • angular motion, circular symmetry

12. Ay Ax 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 • = Ax Ay

13. 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

14. Vector decomposition • Another common decomposition that we’ll use extensively in discussion angular motion uses the radial and tangential directions. at e.g. acceleration along a curved trajectory ar

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

16. A B Vector subtraction • Same as adding negative vector C=A-B A C -B

17. 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

18. Vectors – concept tests

19. 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