Download 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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