Download Mechanics 105 chapter 1

Mechanics 105
Introduction and vectors (chapter one)
Standards
Dimensional analysis
Unit conversions
Order of magnitude calculations/estimation
Significant figures
Coordinates
Vectors and scalars
Problem solving
Mechanics 105
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)
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 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:
E
(vol)
E0

r
And since the scattered irradiance (power/area) is proportional to the square of the
electric field
2
6
I
(vol)
L
4



L
I0
r2
L2
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.
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/T2
3. M/T2
4. It’s dimensionless
Mechanics 105
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?
1.00 inch
57.0 mm 
 2.24 mm
25.4 mm
Units must cancel!
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
x  r  cos( )
y  r  sin(  )
r x y
2
2
tan( )  y / x
y
r

x
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

A  Ax iˆ  Ay ˆj


 A cos iˆ  A sin ˆj
tan  
Ay
Ax

2
2
A  Ax  Ay
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.
at
ar
e.g.
acceleration
along a curved
trajectory
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
A
B
C
B
 
A  B  ( Ax  Bx )iˆ  ( Ay  By ) ˆj
Mechanics 105
Vector subtraction
Same as adding negative vector
A
C=A-B
A
C
B
-B
 
A  B  ( Ax  Bx )iˆ  ( Ay  By ) ˆj
Mechanics 105
Vector multiplication
Mulitplication by a scalar
each component multiplied by scalar

c  A  (c  Ax )iˆ  (c  Ay ) ˆj
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