Force Vectors and Engineering Scales

• Vectors A, B, and C are represented with arrows.

Vector Magnitudes

• Determine the approximate magnitude of vector B in terms of vector A.
• Determine the approximate magnitudes of vector C.
• Vector D is -A.

Engineering Scale

• An engineering scale is used for scaled drawings.

Parallel Lines

• Parallel lines can be drawn using two straight edge triangles.

Scaling Problems

• Use a 1”=30’ scale to draw a horizontal line representing 97’ using an engineering scale of 1:30.
• Use a 1”=200’ scale to draw a vertical line representing 895’ using your engineering scale 1:20.
• Use a 1”=30 lbs scale to draw a force vector going west, representing 97 lbs using your engineering scale 1:30.
• Use a 1”=1000 lbs scale to draw a force vector going Northeast (45o with horizontal) with a magnitude of 4.06 kips=4060 lb, using your engineering scale 1:50.

Adding Force Vectors

• Add force vectors using both learned methods.
• Compare answers using both methods.

Parallelogram method steps

• Using protractor measure 45 and draw a line.
• Measure 5.62/0.1 = 56.2 divisions for vector A.
• Draw parallel lines to form a parallelogram.
• Draw the resultant of A+B.
• Measure BC angle.
• Using protractor measure 12.53 and draw a line for vector.
• Measure 9.219/0.1 = 92.1 divisions.
• Measure and draw 4 = 40 divisions/0.1 for vector C
• Measure 90° and draw a line.
• Draw // lines to (A+B) + C
• measure length of A+B+C = 129 divisions = 12.9 #s

Numerical Example: Method 1

• Select an adequate scale, such that 1” = 2 lb so that the largest number = 9.219# < 5”.
• Using an engineering scale of 1:20, the smallest division is equal to 2/20 = 0.1#.

Method 2 Steps

• Steps to measure and draw lines, angles and vectors until solving for your resultant vector graphically.

Homework

• Redo an example problem.
• Complete a problem within a textbook.

Basic Trigonometry Review:

• α = Arc tan (Opposite/Adjacent) = tan -1 (Opposite/Adjacent)
• α = Arc sin (Opposite/hypotenuse) = sin-1 (Opposite/hypotenuse)
• α = Arc cos (adjacent/hypotenuse) = cos-1 (Adjacent/hypotenuse)
• Pythagoras Theorem: h² = c₁2 + c22

Trigonometry Examples

• Given c = 10 and α =30 degrees; a = c x Sin α = 10 x Sin 30 = 5.
• Cos α = b/c, so b = c x Cos α = 10 x Cos 30 = 8.67
• Given a = 5 and b=5, C = √(a2 + b2) = √(52 + 52) = 7.071, α =Tan -1 (a/b) = Tan -1 (5/5) = 45

2D Vectors

• Fx = F * Cos α
• Fy = F * Sin α
• Tan α = Fy/Fx
• α = tan -1 (Fy/Fx)

Rectangular Components for Force Vectors

• Obtain the rectangular components for force vector F=30 kips onto orthogonal axis X,Y:
  • Fx= F * Cos α = 30 kips*Cos(22.6)=27.7 kips
  • Fy= F * Sin α = 30 * Sin(22.6)= 11.5 kips
  • Check: F=√(Fx2 + Fy2)=√(27.72 + 11.52)= 29.99 = 30 kips
• Obtain the rectangular components for force vector F onto rotated orthogonal axis X’, Y’:
  • F’x= F * Cos α = 30 kips*Cos(65.6)=12.4 kips
  • F'y= F * Sin α = 30 * Sin(65.6)= 27.3 kips
  • Check: F=√(Fx2 + Fy2)=√(12.42 + 27.32)= 29.99 = 30 kips

Tension

• Find the orthogonal components to determine the horizontal component or Tension force and the vertical component or Shear force experienced by an eye screw.
• Fido is pulling with a force of F= 10 lbs

Group Work:

• Read and Discuss problems in groups.

Analytical method

• Assume a sign convention.
• Decompose each vector into its orthogonal components.
• Add the orthogonal components algebraically: ΣFx and ΣFy.
• Determine the angle α of the resultant and the magnitude of the vector.

Forces of Animals

• Add the forces of animals analytically and find the resultant on the steel beam shown.

Textbook example 4

• Solve problem given.
• Complete the exit ticket.

Explore force vectors, their magnitudes, and representation. Practice using engineering scales to draw lines representing real-world measurements and forces, and learn to add force vectors using graphical methods.