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Mechanics
Mechanics
Statics
Dynamics‐Equilibrium
‐Selected Topics
Kinematics Kinetics
‐Particles
‐Rigid Bodies
‐Particles
‐ Rigid Bodies
A branch of physical science which deals with ( the states of rest or motion of ) bodies under action of forces
Dynamics: Motion of bodies
Statics: Equilibrium of bodies (no accelerated motion)under action of Forces
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Mechanics
Mechanics
Statics
Dynamics
Mech. of Materials
Fluid Mechanics
Vibration
Fracture Mechanics
Etc.
Structures
Automotive
Robotics
Spacecraft
Etc.
Basic Concepts
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Basic Concept ‐ Definition
Space: Collection of points whose relative positions can be described using “a coordinate system”Time is a measure of the succession of events and is considered anabsolute quantity.
Mass : is the quantitative measure of the inertia or resistance tochange in motion of a body [Dynamics]. Mass may also be considered as the quantity of matter in a body.
position, velocity, acceleration
r
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Kinematics: is the branch of classical mechanics which describes the motionof points, bodies (objects) and systems of bodies (groups of objects)without consideration of the causes of motion. So you only have velocitiesand accelerations without the forces/torques which creates the motion.
Kinetics: is a term for the branch of classical mechanics that is concernedwith the relationship between the motion of bodies and its causes, namelyforces and torques. So here you have both velocities, accelerations and theforces which creates the motion.
Force: is the vector action of one body on another
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A particle: is a body of negligible dimensions. When the dimensions of abody are irrelevant to the description of its motion or the action of forceson it, the body may be treated as a particle. An airplane, for example, maybe treated as a particle for the description of its flight path.
A rigid body: is a body whose changes in shape are negligible comparedwith the overall dimensions of the body or with the changes in position ofthe body as a whole.
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• In dynamics, force is an action that tends to cause acceleration of an object.
• The SI unit of force magnitude is the newton (N). One newton is equivalent to one kilogram‐meter per second squared (kg∙m/s2 or kg∙m ∙ s – 2)
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SCALARS AND VECTORS
Scalars: associated with “Magnitude” alone
Vectors: associated with “Magnitude” and “Direction”
‐mass, density, volume, time, energy, …
‐ force, displacement, velocity, acceleration, …
: Direction
or V| |V
Magnitude:
Vor V
Vector :
free vector (“math” vector)
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8
Vector: magnitude & direction, components• Scalar multiplication• Addition, subtraction• Dot product• Cross product• Mixed triple product
ManipulationScalar & Vector
,
( )
aAA B A B
A BA B
A B C
Mathematical MeaningsvsPhysical Meanings
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Physical Quantity of VectorVectors representing physical quantities can be classified• Fixed Vector
• Its action is associated with a unique point of application• Described by magnitude, direction & pt of application
• Sliding Vector• Has a unique line of action in space but not a unique point of
application• Described by magnitude, direction & line of action
• Free Vector• Its action is not confined or associated with a unique line in
space.• Described by magnitude & direction
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The Principle of Transmissibility
“A force may be applied at any point on its given line of actionwithout altering the resultant effects external to the rigid bodyon which it acts.”
We can slide the force along its line of action.(force can be considered as sliding vector)
F
F
=?
The two force can be considered equivalent if
……
If we concerns only about the external resultant effects on rigid body.
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Summation of Force
1F
2F
1 2F F
1F
2F
1F 2F
1 2F F
if there are sliding vectors
concurrent forces
non‐concurrent
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NEWTON’S LAWS OF MOTION (1st Law)
The study of rigid body mechanics is formulated on the basis of Newton’s laws of motion.
0F
First Law:An object at rest tends to stay at rest and an object in motion
tends to stay in motion with the same speed and in the same direction, unless acted upon by an unbalanced force.A particle remains at rest or continues to move with uniformvelocity (in a straight line with a constant speed) if there is no unbalancedforce acting on it
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NEWTON’S LAWS OF MOTION (2nd Law)
Second Law:The acceleration of a particle is proportional to the vector sum of
forces acting on it, and is in the direction of this vector sum.
mF
a
amF
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NEWTON’S LAWS OF MOTION
Third Law:The mutual forces of action and reaction between two
particles are equal in magnitude, opposite in direction, and collinear.
F
F
F
F
Confusing? Concept of FBD (Free Body Diagram)
Point: Isolate the body
Forces always occur in pairs – equal and opposite action-reaction force pairs.
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Newton’s Law of Gravitation
2rGMmF
‐M & m are particle masses ‐ G is the universal constant of gravitation,
6.673 x 10‐11 m3/kg‐s2 ‐ r is the distance between the particles.
For Gravity on earth (at sea level)
where ‐ m is the mass of the body in question ‐ g = GM/R2 = 9.81 m/s2 (32.2 ft/s2)
m
M
W=mg
W mg
M
mr F
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Effect of Altitude
g0: represents the absolute acceleration due to gravity at sea level
h :the absolute value at an altitude
R: is the radius of the earth
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Problems pp18
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Kinematics of Particles
Particle motion and Choice of Coordinates
Constrained motion: the particle is confined to a specified path
Unconstrained motion: there are no physical guides
position of particle P at any time t can be described by specifying its rectangularcoordinates* x, y, z, its cylindrical coordinates r, , z,θ or its spherical coordinatesR,θ , φ .Absolute-motion analysis: coordinates measured from fixed reference axes
Relative motion analysis: coordinates measured from moving reference axes
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Rectilinear Motion
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Velocity and Acceleration
then
then
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Acceleration and velocity are vector quantities and can be positive or negative
• Differntial equation relating displcement, velocity and acceleration
• For constant acceleration we can integrate these equations as follows
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SAMPLE PROBLEM 2/1-4 P 27-30
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Visualization of Motion