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)

MassSI: kilogram (kg)esu: gram (g)British: slug

TimeSI: 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: (mass•length/time2)

Energy: (mass•length2/time2)

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=force•L=M L2/T2

Kinetic energy = ½ mv2 = M L2/T2

Mechanics 105

Dimensional analysis continuedAnother (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.

42

6

2

2

0

)(L

L

L

r

vol

I

I

r

vol

E

E )(

0

Mechanics 105

Concept test – dimensional analysis

Hooke’s law for a spring tells us that the force due to a spring is -kx, 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

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.

1.00 inch = 25.4 mm 1.00 = (1.00 inch/25.4 mm)

How many inches is 57.0 mm?

Units must cancel!

Unit conversions

mm 24.2mm 4.25

inch 00.1mm 0.57

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? 1st 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:

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).

Mechanics 105

Coordinate systems

Cartesian (x,y)

linear motion

Polar (r, )

angular motion, circular symmetry

xy

yxr

ry

rx

/)tan(

)sin(

)cos(

22

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

=

Ay

Ax

Ay

Ax

Mechanics 105

Vector decomposition

22

tan

ˆsinˆcos

ˆˆ

yx

x

y

yx

AAA

A

A

jAiA

jAiAA

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

at

ar

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

jBAiBABA yyxxˆ)(ˆ)(

Mechanics 105

Vector subtraction

Same as adding negative vector

A

B

A

-B

C

C=A-B

jBAiBABA yyxxˆ)(ˆ)(

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

jAciAcAc yxˆ)(ˆ)(

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