ENGR 107 – Introduction to Engineering

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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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ENGR 107 - Introduction to Engineering

Coordinate Systems

(in 2 dimensions)

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)
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)
ENGR 107 - Introduction to Engineering

A scalar is a physical quantity that possesses only magnitude.

Scalars and Vectors
ENGR 107 - Introduction to Engineering

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

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?
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.
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

ENGR 107 - Introduction to Engineering

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

ENGR 107 - Introduction to Engineering

Multiplication and Division of Vectors

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

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.
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

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.
ENGR 107 - Introduction to Engineering

The study of forces acting on physical bodies.

Mechanics
ENGR 107 - Introduction to Engineering

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

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.
ENGR 107 - Introduction to EngineeringStatics and Dynamics
• Dynamics is the study of unbalanced forces on a body resulting in an acceleration.
• S F = ma
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
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
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.
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.