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Force Vectors: Notes, Problems and Apps

ID: 1c62cr8keq

Force vectors from start to finish: nine short notes, 160 problems with step-by-step solutions and five interactive apps, from adding forces to the dot product.

H4M Team
H4M TeamAdmin
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  1. 1Section

    Scalars and Vectors

    1 item
    1. 1.1NoteUndergraduateBasic

      Scalars and Vectors in Statics

      What makes force a vector, how vectors are drawn and written, and how free, sliding and fixed vectors differ.
  2. 2Section

    Adding Forces

    22 items · 3 subsections
    1. 2.1NoteUndergraduateBasic

      Adding Forces: Parallelogram Law and Triangle Rule

      Find the resultant of two or more forces with the parallelogram law, the triangle rule and the laws of sines and cosines.
    2. 2.2AppSimulatorHTML5 / JavaScript

      Force Addition Lab: Parallelogram, Triangle and Polygon Rules

      Drag two or more forces and watch their resultant appear by the parallelogram, triangle or polygon rule, with the magnitude and direction worked out live.
    2.3Subsection

    Warm-up

    6 items
    1. 2.3.1ProblemWarm-up

      Resultant of Two Forces 60° Apart

    2. 2.3.2ProblemWarm-up

      Hook on a Wall

    3. 2.3.3ProblemWarm-up

      Pulling a Sled

    4. 2.3.4ProblemWarm-up

      Resultant of Two Perpendicular Forces

    5. 2.3.5ProblemWarm-up

      Difference of Two Forces

    6. 2.3.6ProblemWarm-up

      Pulling Down a Tree

    2.4Subsection

    Workout

    10 items
    1. 2.4.1ProblemWorkout

      Two Tugboats on a Ship

    2. 2.4.2ProblemWorkout

      Two Tractors on a Stuck Truck

    3. 2.4.3ProblemWorkout

      Moving a Pallet Down an Aisle

    4. 2.4.4ProblemWorkout

      Cable Angle for a Required Resultant

    5. 2.4.5ProblemWorkout

      The Missing Force on a Bracket

    6. 2.4.6ProblemWorkout

      Three Ropes, Two Triangles

    7. 2.4.7ProblemWorkout

      How Big Can the Resultant Be?

    8. 2.4.8ProblemWorkout

      Pulling a Car Out of a Ditch

    9. 2.4.9ProblemWorkout

      Second Force for a Vertical Resultant on a Hook

    10. 2.4.10ProblemWorkout

      Two Dogs and a Sled

    2.5Subsection

    Challenge

    4 items
    1. 2.5.1ProblemChallenge

      Pulling a Car Out of a Gap

    2. 2.5.2ProblemChallenge

      Dragging a Log

    3. 2.5.3ProblemChallenge

      Three Winches on a Guide Rail

    4. 2.5.4ProblemChallenge

      Moving a Boulder

  3. 3Section

    Components Along Two Axes

    22 items · 3 subsections
    1. 3.1NoteUndergraduateBasic

      Resolving a Force into Components Along Two Axes

      Split a force into components along any two given directions, and see why oblique components are not projections.
    2. 3.2AppSolverHTML5 / JavaScript

      Force Resolver: Split a Force Along Any Two Directions

      Split a force into two parts along any two directions, perpendicular or not, or into x and y parts from an angle or a slope, with every step shown on the drawing.
    3.3Subsection

    Warm-up

    6 items
    1. 3.3.1ProblemWarm-up

      Force Components on Oblique Axes

    2. 3.3.2ProblemWarm-up

      A Force on the Bisector

    3. 3.3.3ProblemWarm-up

      Perpendicular Axes as a Special Case

    4. 3.3.4ProblemWarm-up

      Components and Projections

    5. 3.3.5ProblemWarm-up

      A Lamp on Two Wires

    6. 3.3.6ProblemWarm-up

      A Wall Bracket

    3.4Subsection

    Workout

    10 items
    1. 3.4.1ProblemWorkout

      Pulling Out a Stump

    2. 3.4.2ProblemWorkout

      Two Cranes, One Panel

    3. 3.4.3ProblemWorkout

      A Sign on Two Chains

    4. 3.4.4ProblemWorkout

      One Component Is Given

    5. 3.4.5ProblemWorkout

      Choosing the Second Rope

    6. 3.4.6ProblemWorkout

      Components Bigger Than the Force

    7. 3.4.7ProblemWorkout

      Crane Jib

    8. 3.4.8ProblemWorkout

      Setting a Component

    9. 3.4.9ProblemWorkout

      Bicycle Frame

    10. 3.4.10ProblemWorkout

      Components Versus Projections

    3.5Subsection

    Challenge

    4 items
    1. 3.5.1ProblemChallenge

      Equal Components Along Two Axes

    2. 3.5.2ProblemChallenge

      One Component Twice the Other

    3. 3.5.3ProblemChallenge

      Felling a Tree Safely

    4. 3.5.4ProblemChallenge

      From Components Back to the Force

  4. 4Section

    Rectangular Components in 2D

    21 items · 3 subsections
    1. 4.1NoteUndergraduateBasic

      Rectangular Components of a Force in 2D

      Write a force by its x and y components, in scalar and Cartesian vector form, and get its magnitude and direction back.
    4.2Subsection

    Warm-up

    6 items
    1. 4.2.1ProblemWarm-up

      A Force in the First Quadrant

    2. 4.2.2ProblemWarm-up

      A Force in the Second Quadrant

    3. 4.2.3ProblemWarm-up

      Direction by a Slope

    4. 4.2.4ProblemWarm-up

      Magnitude and Direction from Components

    5. 4.2.5ProblemWarm-up

      Pulling a Suitcase

    6. 4.2.6ProblemWarm-up

      Components of a Tow Rope Force

    4.3Subsection

    Workout

    10 items
    1. 4.3.1ProblemWorkout

      Weight on an Incline

    2. 4.3.2ProblemWorkout

      Pulling Up a Ramp

    3. 4.3.3ProblemWorkout

      Components from Coordinates

    4. 4.3.4ProblemWorkout

      A Known Horizontal Component

    5. 4.3.5ProblemWorkout

      Magnitude and One Component

    6. 4.3.6ProblemWorkout

      Components on Rotated Axes

    7. 4.3.7ProblemWorkout

      Hydraulic Cylinder

    8. 4.3.8ProblemWorkout

      Wind on a Roof Panel

    9. 4.3.9ProblemWorkout

      Three Ways to Give a Direction

    10. 4.3.10ProblemWorkout

      The Calculator Trap

    4.4Subsection

    Challenge

    4 items
    1. 4.4.1ProblemChallenge

      Choosing the Axes

    2. 4.4.2ProblemChallenge

      From a Component to the Force

    3. 4.4.3ProblemChallenge

      Where to Anchor a Guy Rope

    4. 4.4.4ProblemChallenge

      Components in Two Systems

  5. 5Section

    Resultant of Coplanar Forces

    22 items · 3 subsections
    1. 5.1NoteUndergraduateBasic

      Resultant of Coplanar Forces by Rectangular Components

      Add any number of coplanar forces by summing x and y components, and find an unknown force from a known resultant.
    2. 5.2AppSolverHTML5 / JavaScript

      Coplanar Force Resultant: Table, Polygon and Missing Force

      Add any number of forces in a plane with a clear component table and a force polygon, or find the force that is missing, that balances the others or that turns the resultant to a wanted direction.
    5.3Subsection

    Warm-up

    6 items
    1. 5.3.1ProblemWarm-up

      Two Forces by Components

    2. 5.3.2ProblemWarm-up

      Three Forces on an Eyebolt

    3. 5.3.3ProblemWarm-up

      Three-Way Tug of War

    4. 5.3.4ProblemWarm-up

      Nearly Opposite Forces

    5. 5.3.5ProblemWarm-up

      Adding Cartesian Vectors

    6. 5.3.6ProblemWarm-up

      Wires on a Utility Pole

    5.4Subsection

    Workout

    10 items
    1. 5.4.1ProblemWorkout

      Belt on a Pulley

    2. 5.4.2ProblemWorkout

      Slings on a Crane Hook

    3. 5.4.3ProblemWorkout

      Third Force on a Bracket

    4. 5.4.4ProblemWorkout

      Two Unknown Magnitudes

    5. 5.4.5ProblemWorkout

      Gusset Plate

    6. 5.4.6ProblemWorkout

      Making the Resultant Horizontal

    7. 5.4.7ProblemWorkout

      Direction for a Vertical Resultant

    8. 5.4.8ProblemWorkout

      A Robot Gripper

    9. 5.4.9ProblemWorkout

      Two Cables and a Load

    10. 5.4.10ProblemWorkout

      Third Force on a Ring

    5.5Subsection

    Challenge

    4 items
    1. 5.5.1ProblemChallenge

      The Smallest Resultant of Three Forces

    2. 5.5.2ProblemChallenge

      Two Winches and a Known Pull

    3. 5.5.3ProblemChallenge

      Designing a Member Force

    4. 5.5.4ProblemChallenge

      Two Directions for a Vertical Resultant

  6. 6Section

    Cartesian Vectors in 3D

    22 items · 3 subsections
    1. 6.1NoteUndergraduateBasic

      Cartesian Vectors in 3D: Components and Direction Angles

      Describe a force in 3D with i, j, k components, its magnitude, coordinate direction angles and unit vector.
    2. 6.2AppVisualizerHTML5 / JavaScript

      3D Force Vector Builder: Angles, Two Points and Sums

      Build a force in space from its components, its three direction angles, two angles or two points, turn the 3D view with the mouse, and add several forces into one.
    6.3Subsection

    Warm-up

    6 items
    1. 6.3.1ProblemWarm-up

      Magnitude and Direction Angles

    2. 6.3.2ProblemWarm-up

      The Third Direction Angle

    3. 6.3.3ProblemWarm-up

      Unit Vector of a 3D Vector

    4. 6.3.4ProblemWarm-up

      From Two Angles

    5. 6.3.5ProblemWarm-up

      Drone Thrust

    6. 6.3.6ProblemWarm-up

      A Cable Pulling Down

    6.4Subsection

    Workout

    10 items
    1. 6.4.1ProblemWorkout

      The Missing z Component

    2. 6.4.2ProblemWorkout

      Angles with a Sign

    3. 6.4.3ProblemWorkout

      Pointing an Antenna

    4. 6.4.4ProblemWorkout

      Angles from Components

    5. 6.4.5ProblemWorkout

      A Robot Arm

    6. 6.4.6ProblemWorkout

      Rope on a Flagpole

    7. 6.4.7ProblemWorkout

      Elevation and Bearing

    8. 6.4.8ProblemWorkout

      Can These Angles Exist?

    9. 6.4.9ProblemWorkout

      Writing F = F u

    10. 6.4.10ProblemWorkout

      Mooring Chain of a Buoy

    6.5Subsection

    Challenge

    4 items
    1. 6.5.1ProblemChallenge

      Magnitude from One Component

    2. 6.5.2ProblemChallenge

      Equal Direction Angles

    3. 6.5.3ProblemChallenge

      Half the Force Is Vertical

    4. 6.5.4ProblemChallenge

      A Crane Boom in Space

  7. 7Section

    Resultant of Forces in 3D

    21 items · 3 subsections
    1. 7.1NoteUndergraduateBasic

      Resultant of Concurrent Forces in 3D

      Add concurrent forces in space component by component and find the magnitude and direction angles of the resultant.
    7.2Subsection

    Warm-up

    6 items
    1. 7.2.1ProblemWarm-up

      Resultant of Two Cartesian Forces

    2. 7.2.2ProblemWarm-up

      Direction Angles Plus Components

    3. 7.2.3ProblemWarm-up

      Forces Along the Axes

    4. 7.2.4ProblemWarm-up

      Subtracting Vectors

    5. 7.2.5ProblemWarm-up

      Hook in a Ceiling

    6. 7.2.6ProblemWarm-up

      Two Drones, One Parcel

    7.3Subsection

    Workout

    10 items
    1. 7.3.1ProblemWorkout

      Two Ways to Give a Direction

    2. 7.3.2ProblemWorkout

      Third Force for a Vertical Resultant in 3D

    3. 7.3.3ProblemWorkout

      Guy Wires on a Mast

    4. 7.3.4ProblemWorkout

      Resultant Along an Axis

    5. 7.3.5ProblemWorkout

      Wind and a Stay Cable

    6. 7.3.6ProblemWorkout

      Three Forces, Three Forms

    7. 7.3.7ProblemWorkout

      Two Unknown Pulls and a Vertical Resultant

    8. 7.3.8ProblemWorkout

      Pulling a Tree Three Ways

    9. 7.3.9ProblemWorkout

      The Other Force from a Known Resultant

    10. 7.3.10ProblemWorkout

      Lifting Frame

    7.4Subsection

    Challenge

    4 items
    1. 7.4.1ProblemChallenge

      Turning a Force to Cancel a Component

    2. 7.4.2ProblemChallenge

      Holding Down a Balloon

    3. 7.4.3ProblemChallenge

      Smallest Force for a Vertical Resultant

    4. 7.4.4ProblemChallenge

      Choosing Two Equal Forces

  8. 8Section

    Force Along a Line

    21 items · 3 subsections
    1. 8.1NoteUndergraduateBasic

      Force Vector Directed Along a Line Using Position Vectors

      Write a cable or rope force as a Cartesian vector from the coordinates of two points, using position and unit vectors.
    8.2Subsection

    Warm-up

    6 items
    1. 8.2.1ProblemWarm-up

      Head Minus Tail

    2. 8.2.2ProblemWarm-up

      A Force Toward a Point

    3. 8.2.3ProblemWarm-up

      Guy Wire on a Mast

    4. 8.2.4ProblemWarm-up

      Unit Vector Between Two Points

    5. 8.2.5ProblemWarm-up

      Tent Pole

    6. 8.2.6ProblemWarm-up

      Direction Angles of a Cable

    8.3Subsection

    Workout

    10 items
    1. 8.3.1ProblemWorkout

      Two Ropes on a Flagpole

    2. 8.3.2ProblemWorkout

      A Plate on Three Cables

    3. 8.3.3ProblemWorkout

      No Sideways Pull

    4. 8.3.4ProblemWorkout

      A Brace That Pushes

    5. 8.3.5ProblemWorkout

      Rope Across a Room

    6. 8.3.6ProblemWorkout

      Two Cables from One Point

    7. 8.3.7ProblemWorkout

      From Points to Two Angles

    8. 8.3.8ProblemWorkout

      Awning Cable

    9. 8.3.9ProblemWorkout

      Which Way Does the Mast Lean?

    10. 8.3.10ProblemWorkout

      Tension from One Component

    8.4Subsection

    Challenge

    4 items
    1. 8.4.1ProblemChallenge

      Balancing Three Guy Wires

    2. 8.4.2ProblemChallenge

      Placing an Anchor

    3. 8.4.3ProblemChallenge

      Tensions for a Vertical Lift

    4. 8.4.4ProblemChallenge

      Keeping the Resultant in a Plane

  9. 9Section

    Dot Product and Projection

    22 items · 3 subsections
    1. 9.1NoteUndergraduateAdvanced

      Dot Product: Angle Between Vectors and Projection of a Force

      Use the dot product to find the angle between two lines and the parallel and perpendicular components of a force.
    2. 9.2AppVisualizerHTML5 / JavaScript

      Dot Product Visualizer: Angles and Projections in 3D

      See what the dot product means: the angle between two directions and the part of a force along a line, drawn in 3D with its perpendicular part and a right-angle mark.
    9.3Subsection

    Warm-up

    6 items
    1. 9.3.1ProblemWarm-up

      Dot Product and Angle

    2. 9.3.2ProblemWarm-up

      Two Edges of a Box

    3. 9.3.3ProblemWarm-up

      Projections of a Force on Two Lines

    4. 9.3.4ProblemWarm-up

      Perpendicular or Not?

    5. 9.3.5ProblemWarm-up

      Cart on a Rail

    6. 9.3.6ProblemWarm-up

      Angle Between Two Cables from Coordinates

    9.4Subsection

    Workout

    10 items
    1. 9.4.1ProblemWorkout

      Cable on a Pipe

    2. 9.4.2ProblemWorkout

      Trap Door Hinge

    3. 9.4.3ProblemWorkout

      Space Frame Members

    4. 9.4.4ProblemWorkout

      Cable Force Along a Beam

    5. 9.4.5ProblemWorkout

      Wind on a Solar Panel

    6. 9.4.6ProblemWorkout

      Cable and Pole

    7. 9.4.7ProblemWorkout

      Projection on a Line Through Two Points

    8. 9.4.8ProblemWorkout

      Parallel and Perpendicular Components of a Force

    9. 9.4.9ProblemWorkout

      Angle with the Floor

    10. 9.4.10ProblemWorkout

      Resultant from the Dot Product

    9.5Subsection

    Challenge

    4 items
    1. 9.5.1ProblemChallenge

      Making a Right Angle

    2. 9.5.2ProblemChallenge

      Smallest Force for a Resultant Along a Line

    3. 9.5.3ProblemChallenge

      Where to Anchor the Cable

    4. 9.5.4ProblemChallenge

      Limiting the Axial Pull

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