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Objectives

  1. To show how to add forces and resolve them into components using the parallelogram law.
  2. To express force and position in Cartesian vector form and explain how to determine the vector’s magnitude and direction.

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Definitions

Scalar - A quantity characterized by a positive or negative number is called a scalar. Examples of scalars used in Statics are mass, volume or length.

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Definitions

Vector - A quantity that has both magnitude and a direction. Examples of vectors used in Statics are position, force, and moment.

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Symbols

Vectors are denoted by a letter with an arrow over it or a boldface letter such as A.

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Vector Definitions

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Magnitude and Multiplication of Vector by Scalar

  • The magnitude of a quantity is always positive.
  • If m is scalar quantity and it z multiplied to a vector A we get mA.
  • What does it mean?

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  • mA is vector having same direction as A and magnitude equal to the ordinary scalar product between the magnitude of m and A.

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  • what happens if m is negative?

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Scalar Multiplication

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Scalar Multiplication

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Vector Addition

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  • Vector addition is commutative and associative.

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  • How?

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Vector Addition

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Vector Addition

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Vector Subtraction

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Resolution of a Vector

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Trigonometry

a

b

c

C

B

A

Law of Sines:

Law of Cosines

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Force

  1. Force is a Vector Quantity
  2. Forces Add as Vectors

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Parallelogram Law

  1. Make a sketch showing vector addition using the parallelogram law.
  2. Determine the interior angles of the parallelogram from the geometry of the problem.
  3. Label all known and unknown angles and forces in the sketch.
  4. Redraw one half of the parallelogram to show the triangular head-to-tail addition of the components and apply laws of sines and cosines.

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Important Points

  1. A scalar is a positive or negative number.
  2. A vector is a quantity that has magnitude, direction, and sense.
  3. Multiplication or division of a vector by a scalar will change the magnitude. The sense will change if the scalar is negative.
  4. If the vectors are collinear, the resultant is formed by algebraic or scalar addition.

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Example

The screw eye in the figure at the left is subjected to two forces F1 and F2. Determine the magnitude and direction of the resultant force.

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Parallelogram� Law

Calculate angles

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Angle COA = 900 -150-100 = 650

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Angle OAB = 1800 -650= 1150

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Triangular Construction

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Find FR from law of cosines.

Find θ from law of sines.

Angle φ = θ + 150

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Answer

The resultant force has a magnitude of 213 N and is directed 54.8o from the horizontal.

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Example

Resolve the 200 lb force into components in the x and y directions and in the x’ and y directions

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x

y

x’

y’

300

400

F=200 lb

Resolve the 200 lb force into components in the x and y directions and in the x’ and y directions

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Parallelogram� Law

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Triangular Construction

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Solution – Part (a)

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Parallelogram� Law

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200 lb

Fx’

Fy

300

600

500

400

500

x’

y

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Triangular Construction

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200 lb

Fx’

Fy

600

700

500

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Addition of a System of Coplanar Forces

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Addition of a System of Coplanar Forces

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Cartesian Notation

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Cartesian Notation

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Coplanar Force Resultants

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Resolve into Cartesian Components

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Add Components

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5

12

13

3

4

5

Special Triangles

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3

4

5

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5

12

13

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Example

The link in the figure is subjected to two forces, F1 and F2. Determine the resultant magnitude and orientation of the resultant force.

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Scalar Solution

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Scalar Solution

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Cartesian Vector Solution

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Cartesian Vector Solution