caps-textbook-physical-science-grade11__1_

Vectors and Resultant Forces

Resultant Vector Calculation

  • Forces F1 and F2:

    • F1: 5.00 N (x-component), 2.00 N (y-component), Total: 5.385 N

    • F2: 3.00 N (x-component), 4.00 N (y-component), Total: 5 N

    • Resultant Forces:

      • Rx = 8.00 N

      • Ry = 6.00 N

Magnitude and Direction of Resultant Vector

  • Use Pythagorean Theorem:

    • R² = Rx² + Ry²

    • R² = (8.00)² + (6.00)² = 100.00

    • R = 10.00 N

  • Direction Calculation:

    • Angle α from positive x-axis:

      • tan(α) = Ry / Rx

      • α = tan⁻¹(6.00 / 8.00) = 36.9°

  • Final Result:

    • R = 10.00 N at an angle of 36.9° to the positive x-axis.

Worked Example 15: Resultant Forces from Components

Problem: Resolve Four Forces

  • Given Forces:

    • F1 = 3.5 N at 45°

    • F2 = 2.7 N at 63°

    • F3 = 1.3 N at 127°

    • F4 = 2.5 N at 245°

Step 1: Sketch the Problem

  • Draw vectors on the Cartesian plane with correct positioning.

Step 2: Determine Components

  • F1 Components:

    • F1y = F1 * sin(45°) = 2.47 N

    • F1x = F1 * cos(45°) = 2.47 N

  • F2 Components:

    • F2y = 2.7 * sin(63°) ≈ 2.41 N

    • F2x = 2.7 * cos(63°) ≈ 1.23 N

  • F3 Components:

    • F3y = 1.3 * sin(127°) ≈ 1.04 N

    • F3x = 1.3 * cos(127°) ≈ -0.78 N

  • F4 Components:

    • F4y = 2.5 * sin(245°) ≈ -2.27 N

    • F4x = 2.5 * cos(245°) ≈ -1.06 N

Step 3: Sum the Components

  • Resultant Components:

  • Rx = F1x + F2x + F3x + F4x

  • Ry = F1y + F2y + F3y + F4y

Calculations for Resultant Forces

  • Summed Results:

    • Rx = 1.86 N

    • Ry = 3.65 N

Step 4: Resultant Magnitude and Direction

  • Magnitude:

    • R² = Rx² + Ry² = (1.86)² + (3.65)² = 16.78

    • R = 4.10 N

  • Direction:

    • Angle Calculation: tan(α) = 1.86 / 3.65

    • α ≈ tan⁻¹(3.65/1.86) = 27.00°

  • Final Result:

    • Resultant = 4.10 N at 27.00° to positive x-direction.

Apparatus for Experiment

Informal Experiment: Force Board

  • Aim: Determine resultant of three non-linear forces.

  • Materials Required:

    • Blank paper

    • Force board

    • Spring balances

    • Assorted weights

    • String

    • Pulleys

Method

  1. Set up forceboard with paper underneath.

  2. Connect weights to the ring using spring balances.

  3. Draw lines along each cord on paper and note spring balance readings.

  4. Draw lines representing forces on paper back to ring's centre.

  5. Use scale to draw arrows corresponding to spring balance readings.

Observations

  • Explore the direction and magnitude of resultant from various force vector combinations.

Chapter Summary for Vectors

  • A vector has both magnitude and direction.

  • Used to represent physical quantities like forces.

  • Vectors can be added graphically or algebraically.

  • Components of vectors can be determined using trigonometric functions based on the angle with the x-axis:

    • Rx = R * cos(θ)

    • Ry = R * sin(θ).

  • Graphical representation is done using head-to-tail or parallelogram methods.