1.1
1 Introduction - Mechanics
Mechanics: Science that describes and predicts the
conditions of rest or motion of bodies under the
action of forces. There are multiple branches of
mechanics:
Mechanics of rigid bodies
Mechanics of deformable bodies
Mechanics of incompressible/compressible fluids
and gases
Relativistic and quantum mechanics
The simplest branch – rigid body mechanics (RBM)
consists of:
Statics – considers bodies in equilibrium: being
at rest or at constant velocity
Dynamics – considers bodies in the state of
accelerated motion
It is possible to consider every problem of RBM as a
problem of dynamics, but it will be unnecessarily
difficult.
1.2
Statics
Statics is a foundation of mechanics and strong
knowledge of statics is required to study dynamics
and mechanics of materials.
Basic quantities within the framework of
mechanics:
Space – describes position of a point in space and
geometric properties of bodies (size, shape, etc.)
Time – describes succession of events
Mass – measures resistance of bodies to a change
in velocity (=acceleration)
Force – describes action of one body on another. It
is a vector quantity. Distinguished as contact or
volumetric
In Newtonian mechanics, space, time, and mass are
independent of each other (not so in relativistic
mechanics). Force is not independent and is related
to the body’s mass and the way its velocity changes
with time.
1.3
The Parallelogram Law for the addition of forces.
Two forces acting on a particle may be replaced by a
single force, called the resultant.
http://tutorial.math.lamar.edu
/Classes/CalcII/VectorArithm
etic.aspx
The Principle of Transmissibility. Conditions of
equilibrium or of motion of rigid body will remain
unchanged if a force acting at a given point of the
rigid body is replaced by a force of the same
magnitude and direction, but acting at a different
point, provided the two forces have the same line of
action.
http://yesitsengineering.blogspot.
com/2013_08_01_archive.html
1.4
Newton’s Laws
First Law. A particle originally at rest or moving in a
straight line with constant velocity, will remain in this
state, provided the particle is not subjected to an
unbalanced force.
Second Law. A particle acted upon by an
unbalanced force F experiences an acceleration a
that has the same direction as the force and a
magnitude that is directly proportional to the force.
For the particle of mass m: F=ma.
https://www.haikudeck.com/
newtons-laws-of-motionjacie-andrews-educationpresentationXkexDkN30k#slide-5
Third Law. The mutual forces of action and reaction
between two particles are equal, opposite, and
collinear.
1.5
Statics vs. Dynamics
Time independent – principles of statics are time
independent and acceleration is not present. Hence,
only first and third laws are important for statics.
Newton’s law of Gravitation:
http://philschatz.com/physicsbook/contents/m42143.html
𝑀𝑚
𝐹=𝐺 2
𝑟
Where r is the distance between the bodies of
masses M and m. G is a universal constant called
constant of gravitation.
Weight of the particle (W). Gravitational force F
exerted by Earth on a particle of mass m.
𝑊 = 𝑚𝑔
Where
1.6
𝑀
𝑚
𝑓𝑡
𝑔 = 𝐺 2 ~9.81 2 = 32.2 2
𝑅
𝑠
𝑠
Idealizations
Particle is a mass of negligible size. Allows
maximum simplifications, when the shape and size
of the body don’t matter.
Rigid Body is a body of fixed, underformable shape.
Allows substantial simplifications, when changes of
shape of the body don’t matter. Precise calculation
of real-world deformations is complex.
Concentrated force is a load acting at a point as
opposed to a distributed force.
Examples: Earth, bike, tennis ball hits racquet.
Note: Every idealization should be evaluated, since
one level of idealization can be appropriate for the
same body in one case and not appropriate in
another case.
1.7
Units
Every mechanical quantity or an algebraic
combination of mechanical quantities has units.
Carrying out units throughout calculations is
important:
• Incorrect units are often related to incorrect
answer (but not necessarily vice versa).
Typically, units-related error is very large.
• Tracing incorrect units might help with finding
error – at certain step there can be a mismatch
between units – lack of homogeneity
(guesswork in choosing equations, adding or
missing equation’s elements).
Note: Incorrect units are considered an error,
despite having a correct answer.
1.8
Systems of Units
SI (Systeme International) = Metric = International:
• Basic units: L(meter), T(second), M(kilogram).
• Derived units:
F(newton=𝑁 =
𝑘𝑔∗𝑚
𝑠2
).
Note: W = mg, with m usually given in kg.
U.S. Customary = British = FPS (foot-poundsecond):
• Basic units: L(foot), T(second), F(pound).
• Derived units:
M (slug=
𝑙𝑏∗𝑠 2
𝑓𝑡
).
Note: When force is given in pounds there is no
need to multiply it by g.
1.9
SI Prefixes and Units Used in Mechanics
1.10
Unit Conversion
1.11
Numerical vs. Symbolic
Numerical solution: Answer is found of a number
and at times it seems easier to calculate it with the
use of a calculator. However, it will require consistent
check for units’ homogeneity - all terms should have
same units/dimensions and it should be checked
before crunching numbers. Numerical answer is
problem specific and subject to accuracy of results
being lower than accuracy of data due to a round-off.
Symbolic solution: Answer is found in form of a
formula with much deeper perception of
dependencies. Since the symbolic answer is general
it is possible to re-use it, unlike the numerical
answer.
Generally symbolic is more powerful!
1.12
General Procedure for Analysis
• Read the problem. Make sure to correlate every
given fact to studied, while not adding your own
facts (anecdotally!).
• Draw necessary diagrams.
• Apply relevant principles in mathematical form.
• Solve necessary equations algebraically as far as
practical, then, making sure they are dimensionally
homogeneous, use a consistent set of units and
complete solutions numerically. Report the answer
with no more significant figures than the accuracy
of the given data.
• Study the answer with technical judgment and
common sense to determine whether or not it
seems reasonable.
• Review the solution and think of other ways to
obtain it.
• Try to solve problem as neatly as possible – it
helps to see general picture and to avoid regretful
mistakes.
1.13
Numerical Accuracy
The accuracy of the solution depends on two items:
1. Accuracy of data
2. Accuracy of computations
The solution can’t be more accurate than the less
accurate of the two items.
Example: Loading of a structure is known to be
75,000 lb with a possible error of 100 lb.
Relative error:
100 𝑙𝑏
= 0.0013 = 0.13%
75,000 𝑙𝑏
Which means that the accuracy in solution (such as
computing reaction in supports) can’t be greater
than 0.13%. If the computation gives 14,322 lb, the
possible error is (0.13% of 14,322~20 lb), which
means that there is no point in recording all the
digits from the computation. Rather the answer
should be recorded as 14,320±20 lb.
Note: In most engineering problems the data is
seldom known to accuracy greater than 0.2%