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Coordinate Systems, Vectors, and Forces (Lecture #6). ENGR 107 – Introduction to Engineering. Coordinate Systems (in 2 dimensions). 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)

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slide2
ENGR 107 - Introduction to Engineering

Coordinate Systems

(in 2 dimensions)

coordinate systems
ENGR 107 - Introduction to EngineeringCoordinate 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)
cartesian polar
ENGR 107 - Introduction to EngineeringCartesian ↔ 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)
a scalar is a physical quantity that possesses only magnitude
ENGR 107 - Introduction to Engineering

A scalar is a physical quantity that possesses only magnitude.

Scalars and Vectors
a vector is a physical quantity that possesses both magnitude and direction
ENGR 107 - Introduction to Engineering

A vector is a physical quantity that possesses both magnitude and direction.

Scalars and Vectors
scalars and vectors
ENGR 107 - Introduction to EngineeringScalars and Vectors
  • Which are scalars and which are vectors?

Time Acceleration

Force Speed

Distance Temperature

Mass Velocity

  • Other examples?
vectors
ENGR 107 - Introduction to EngineeringVectors
  • 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 y-directions, respectively,
    • And i and j are the unit vectors in the x- and y-directions, respectively.
vectors1
ENGR 107 - Introduction to EngineeringVectors
  • 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

slide16
ENGR 107 - Introduction to Engineering

Addition and Subtraction of Vectors

addition and subtraction
ENGR 107 - Introduction to EngineeringAddition 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

slide19
ENGR 107 - Introduction to Engineering

Multiplication and Division of Vectors

addition and subtraction1
ENGR 107 - Introduction to EngineeringAddition 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

forces
ENGR 107 - Introduction to EngineeringForces
  • 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.
forces1
ENGR 107 - Introduction to Engineering

y

F.cosq

F

FY

F.sinq

q

x

FX

Forces

F = |F| < q

F = FXi + FYj

Fx = F.cosq

Fy = F.sinq

forces2
ENGR 107 - Introduction to EngineeringForces
  • The force, F, can be resolved into its two vector components, FX and FY.
    • FX = F.cosqi
    • FY = F.sinqj
  • 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.
the study of forces acting on physical bodies
ENGR 107 - Introduction to Engineering

The study of forces acting on physical bodies.

Mechanics
branches of mechanics concerned with the analysis of forces on rigid bodies
ENGR 107 - Introduction to Engineering

Branches of mechanics concerned with the analysis of forces on rigid bodies.

Statics and Dynamics
statics and dynamics
ENGR 107 - Introduction to EngineeringStatics 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.
    • S F = 0
    • The body is in static equilibrium.
statics and dynamics1
ENGR 107 - Introduction to EngineeringStatics and Dynamics
  • Dynamics is the study of unbalanced forces on a body resulting in an acceleration.
    • S F = ma
static equilibrium
ENGR 107 - Introduction to EngineeringStatic 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
slide31
ENGR 107 - Introduction to Engineering

To implement the analysis of a rigid body in static equilibrium, one must first draw a

Free Body Diagram (FBD).

Statics
free body diagrams
ENGR 107 - Introduction to EngineeringFree-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.
free body diagram
ENGR 107 - Introduction to EngineeringFree-Body Diagram

Steps for drawing a FBD:

  • Isolate the desired object from its surroundings.
  • Replace items cut free with appropriate forces.
  • Add known forces, including weight.
  • Establish a coordinate (xy) frame of reference.
  • Add geometric data.
examples to include only analysis of forces moments will be discussed later
ENGR 107 - Introduction to Engineering

Examples

To include only analysis of forces.

Moments will be discussed later.

Statics