ENGR 107 – Introduction to Engineering
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Transcript ENGR 107 – Introduction to Engineering
ENGR 107 – Introduction to Engineering
Coordinate Systems,
Vectors,
and
Forces
(Lecture #6)
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Coordinate Systems
(in 2 dimensions)
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Coordinate Systems
Cartesian Coordinate System
Each point in the plane is specified by the
perpendicular distance to the x-, and y- axes.
P(x, y)
Polar Coordinate System
Each point in the plane is specified by the
radial distance from the pole (or origin) and
the angle to the horizontal axis.
P(r, q)
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Cartesian Coordinate System
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Cartesian Coordinate System
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Polar Coordinate System
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Polar Coordinate System
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Cartesian ↔ Polar
For a point P specified in the
Cartesian Coordinate System:
P(x, y)
Polar Coordinate System:
P(r, q)
r2 = x2 + y2 → r = sqrt[ x2 + y2 ]
q = arctan( y / x )
x = r.cos(q)
y = r.sin(q)
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Cartesian ↔ Polar
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Scalars and Vectors
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Scalars and Vectors
A scalar is a physical quantity that possesses
only magnitude.
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Scalars and Vectors
A vector is a physical quantity that possesses
both magnitude and direction.
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Scalars and Vectors
Which are scalars and which are vectors?
Time
Acceleration
Force
Speed
Distance
Temperature
Mass
Velocity
Other examples?
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Vectors
In the Cartesian Coordinate System
A = AXi + AYj
where A is the vector quantity,
AX and AY are the magnitudes of the
rectangular components in the x- and ydirections, respectively,
And i and j are the unit vectors in the x- and ydirections, respectively.
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Vectors
In the Polar Coordinate System
A=A<q
where A is the vector quantity,
A is the magnitude (a scalar quantity)
and q is the angle (with respect to the x-axis)
note: A = |A| = magnitude of A
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Addition and Subtraction of Vectors
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Addition and Subtraction
Vectors should be written in rectangular form.
Cannot add or subtract vectors directly when
written in polar form.
Add the x- and y- components independently.
R=A+B
Rx = Ax + Bx
Ry = Ay + By
R = Rxi + Ryj
A = Axi + Ayj
B = Bxi + Byj
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Addition and Subtraction
Exercises
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Multiplication and Division of Vectors
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Addition and Subtraction
Vectors should be written in polar form.
More difficult to multiply and divide vectors
when written in rectangular form.
Multiply the magnitudes and add the angles.
R=A.B
R=A.B
qR = qA + qB
R = R < qR
A = A < qA
B = B < qB
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Multiplication and Division
Exercises
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Forces
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Forces
A force is an action, a push or a pull, that tends to
change the motion of the body acted upon.
A force has both magnitude and direction
Thus, it is a vector.
A force may be moved along its line of action
without altering the external effect.
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Forces
y
F = |F| < q
F.cosq
F
F = F Xi + F Yj
FY
F.sinq
q
FX
Fx = F.cosq
Fy = F.sinq
x
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Forces
The force, F, can be resolved into its two vector
components, FX and FY.
FX = F.cosq i
FY = F.sinq j
The combined effect of the vector components of
a force, FX and FY, applied to a body is equivalent
to the net effect of the force F applied to the
body.
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Mechanics
The study of forces acting on physical bodies.
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Statics and Dynamics
Branches of mechanics concerned with the
analysis of forces on rigid bodies.
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Statics and Dynamics
Statics is the study of balanced forces on a body
resulting in the body remaining at rest or moving
with a constant velocity.
SF=0
The body is in static equilibrium.
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Statics and Dynamics
Dynamics is the study of unbalanced forces on a
body resulting in an acceleration.
S F = ma
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Static Equilibrium
A body will be in static equilibrium when the sum
of all external forces and moments acting on the
body is zero.
Conditions of static equilibrium:
S FX = 0
S FY = 0
S MP = 0
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Statics
To implement the analysis of a rigid body in
static equilibrium, one must first draw a
Free Body Diagram (FBD).
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Free-Body Diagrams
A Free-Body Diagram (FBD) is a sketch of the
body, or a portion of the body, and all of the
forces acting upon the body.
The body is “cut free” from all others, and only
forces that act upon it are considered.
Must have an understanding of the types of
reactions that may occur at supports and
connectors.
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Free-Body Diagram
Steps for drawing a FBD:
1. Isolate the desired object from its surroundings.
2. Replace items cut free with appropriate forces.
3. Add known forces, including weight.
4. Establish a coordinate (xy) frame of reference.
5. Add geometric data.
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Free Body Diagram
Examples
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Statics
Examples
To include only analysis of forces.
Moments will be discussed later.
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