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

Mechanics

Mechanics

Statics

Dynamics

Mech. of Materials

Fluid Mechanics

Vibration

Fracture Mechanics

Etc.

Structures

Automotive

Robotics

Spacecraft

Etc.

Basic Concepts

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

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

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.

• 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)

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)

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

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

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.

Summation of Force 

1F

2F

1 2F F

1F

2F

1F 2F

1 2F F

if there are sliding vectors

concurrent forces

non‐concurrent

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

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

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.

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

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

Problems pp18

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

Rectilinear Motion

Velocity and Acceleration                         

then

then

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

SAMPLE PROBLEM 2/1-4 P 27-30

Visualization of Motion