Finding Resultant Force of Coplanar Forces
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Questions and Answers

What is the resultant of the five coplanar forces acting on the object as shown in the figure?

  • 4.9 N
  • 3.6 N
  • 6.5 N
  • 2.5 N (correct)
  • 5.6 N

Determine the angle between vectors $\vec{A} = -2\hat{i} + 3\hat{j}$ and $\vec{B} = 2\hat{i} - 6\hat{j}$

  • 105°
  • 165°
  • 30°
  • 145° (correct)
  • 15°

Find the resultant of the five coplanar forces acting on an object as shown in the attached figure.

  • 6.5 N
  • 2.5 N (correct)
  • 5.6 N
  • 4.9 N
  • 3.6 N

Flashcards

Coplanar Forces

Forces that lie on the same plane.

Resultant Force

The single force that produces the same effect as all the original forces combined.

Resolving Forces

Break each force into x and y components using trigonometry (sine and cosine).

Component Calculation Formulas

Fx = F * cos(θ), Fy = F * sin(θ), where θ is measured from the positive x-axis.

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Sign Conventions for Components

Pay attention to the quadrant of the angle to determine the correct sign (+ or -) for each component.

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Summing Components

Sum all the x-components (ΣFx) and sum all the y-components (ΣFy).

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Magnitude of Resultant Force

Use Pythagorean theorem: R = √(ΣFx² + ΣFy²)

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Direction of Resultant Force

Use the inverse tangent function: θ = tan⁻¹(ΣFy / ΣFx)

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Standard Angle Convention

The angle is measured counter-clockwise from the positive x-axis.

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Calculator Mode

Ensure your calculator is in degree mode when calculating trigonometric functions.

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Dot Product Definition

A dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number.

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Dot Product Formula (using angle)

A ⋅ B = |A| |B| cos(θ), where θ is the angle between vectors A and B.

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Dot Product Formula (components)

A ⋅ B = AxBx + AyBy + AzBz (sum of products of corresponding components).

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Angle Between Vectors Formula

cos(θ) = (A ⋅ B) / (|A| |B|)

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Vector Magnitude

The length or magnitude of vector A is denoted as |A| and is calculated as √(Ax² + Ay² + Az²).

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Component Identification

For A = -2i + 3j, the components are Ax = -2 and Ay = 3.

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Dot Product Calculation (Step 1)

First, find the dot product of vectors A and B using their components.

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Magnitude Calculation (Step 2)

Next, calculate the magnitudes of vectors A and B (|A| and |B|).

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Cosine Calculation (Step 3)

Use the values from the previous steps in cos(θ) = (A ⋅ B) / (|A| |B|) to find the cosine of the angle.

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Angle Calculation (Step 4)

θ = arccos( (A ⋅ B) / (|A| |B|) )

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Study Notes

  • The objective is to find the resultant force of five coplanar forces acting on an object.
  • Coplanar forces mean all forces lie in the same plane.

Force vectors breakdown

  • Five forces are acting on the object: 16 N, 15 N, 19 N, 11 N, and 22 N.
  • The 16 N force acts at an angle of 45° relative to the negative x-axis.
  • The 15 N force acts at an angle of 60° relative to the positive y-axis.
  • The 19 N force acts along the positive x-axis.
  • The 11 N force acts at an angle of 30° relative to the negative x-axis.
  • The 22 N force acts along the negative y-axis.

Calculating Resultant Force

  • Break down each force into x and y components.
  • Add all x-components to get the resultant x-component.
  • Add all y-components to get the resultant y-component.
  • Use the Pythagorean theorem to find the magnitude of the resultant force.
  • Use trigonometry to find the direction of the resultant force.

X-Component Calculations

  • 16 N force: -16 * cos(45°) = -11.31 N
  • 15 N force: 15 * sin(60°) = 12.99 N
  • 19 N force: 19 N
  • 11 N force: -11 * cos(30°) = -9.53 N
  • 22 N force: 0 N
  • Resultant x-component: -11.31 + 12.99 + 19 - 9.53 + 0 = 11.15 N

Y-Component Calculations

  • 16 N force: 16 * sin(45°) = 11.31 N
  • 15 N force: 15 * cos(60°) = 7.5 N
  • 19 N force: 0 N
  • 11 N force: -11 * sin(30°) = -5.5 N
  • 22 N force: -22 N
  • Resultant y-component: 11.31 + 7.5 + 0 - 5.5 - 22 = -8.69 N

Magnitude of Resultant Force

  • Use the Pythagorean theorem: R = sqrt((11.15)^2 + (-8.69)^2)
  • R = sqrt(124.32 + 75.52) = sqrt(199.84) ≈ 14.14 N

Resultant

  • The provided possible answers do not match the calculated resultant, which is approximately 14.14 N.

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Description

Learn how to find the resultant force of multiple coplanar forces acting on an object. This involves breaking down each force into x and y components, adding the components, and then using the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant force.

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