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Kinetics of Particles
 Free Body Diagrams
 Newton’s Laws
 Euler’s Laws
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
1
Free Body Disgrams
 The basic, graphical, force accounting
tool for the analyst
 Drawing FREE BODY DIAGRAMS
 A right handed Cartesian Coordinate system
must be attached to a point on the structure
 All geometric points necessary to a force
analysis must be clearly marked
 Triplet of ordered numbers referenced to
the coordinate system
 Dimensions referenced from the origin
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
2
Free Body Diagrams Continued
 Unit Vectors MUST be shown in the
figure
 Isolate the structure or element from
it’s surroundings
 Sketch all forces that act on the
structure or element
 Known forces should be labeled with their
proper magnitude and direction
 Unknown forces magnitude and direction
will be assumed
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
3
Newton’s Laws of Motion
 Basis for entire subject of rigid
body mechanics
 Based on experimental
observation
 Apply to the motion of particles as
measured from a non-accelerating
reference frame
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
4
Newton’s FIRST LAW
A particle remains at rest or continues
to move in a straight line with uniform
velocity if there is no unbalanced force
acting on it.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
5
Newton’s SECOND LAW
The acceleration of a particle is
propositional to the resultant force
acting on it and is in the direction of
the resultant force


F  ma
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
6
Newton’s THIRD LAW
The forces of action and reaction
between interacting bodies are
equal in magnitude, opposite in
direction, and collinear.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
7
Euler’s Laws
 Important extension of Newton’s
laws
 Restated laws in terms of force,
mass, and absolute acceleration
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
8
Euler’s FIRST LAW
The resultant of the external
forces on a body is at all times
equal to the time derivative of its
momentum
 d

 F  dt m  v 
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
9
Euler’s SECOND LAW
The resultant moment of these
external forces about a fixed point
is equal to the time derivative of
the body’s moment of momentum
about the point.
 
 d

M

I



dt
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
10
Archimedean Versus Newtonian
Equilibrium
 Archimede’s concept of Equilibrium
 The fundamental principle of physics that applies
to all force systems that act on bodies that are at
rest, remain at rest, relative to the surface of the
earth
 Newon’s concept of Equilibrium
 A body is in equilibrium only if it is at rest relative
to the “Fixed Set of Start” or it travels at a
constant speed
 This course is based on the Archimedean
concept of equilibrium.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
11
Example 1
The 10lb block has an
initial velocity of
10ft/s on the smooth
plane. If a force
F=(2.5t) lb, where t is
in seconds, acts on
the block for 3s,
determine the final
velocity of the block
and the distance the
block travels during
this time.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
12
Example 2
A particle falls under the force of gravity in
a medium that exerts a resisting force
proportional to the velocity of the particle.
Develop equations for the velocity and
displacement of the particle. The velocity
and displacement of the particle are zero at
time t=0.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
13
Example 3
Each of the two
blocks has a mass m.
The coefficient of
kinetic friction at all
surfaces of contact is
μ. If a horizontal force
P moves the bottom
block, determine the
acceleration of the
bottom block in each
case
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
14
Example 4
A conical pendulum
consists of a 10 lb sphere
supported by a 6ft cord
that revolves about a
vertical axis with a
constant angular velocity
so that the string is
inclined 30° to the
vertical. Determine the
tension T in the string and
the linear velocity of the
sphere.
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
15
Free Body – Mass Acceleration
Diagrams
Union College
Mechanical Engineering
ESC020: Rigid Body Mechanics
16