Two forces act at a gusset plate: 44 kN at 22° and 81 kN at 139° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$44cos22^{\circ} = 40.80 kN, 44sin22^{\circ} = 16.48 kN$$

3. F₂ components

   $$81cos139^{\circ} = -61.13 kN, 81sin139^{\circ} = 53.14 kN$$

4. Sums — ΣFₓ = -20.34 kN, ΣF_y = 69.62 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-20.34^{2} + 69.62^{2}) = 72.53 kN at 106.3^{\circ}$$

Why the other options are there

• 125 kN (magnitudes added)
• 20.34 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 69 kN at 20° and 83 kN at 130° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$69cos20^{\circ} = 64.84 kN, 69sin20^{\circ} = 23.60 kN$$

3. F₂ components

   $$83cos130^{\circ} = -53.35 kN, 83sin130^{\circ} = 63.58 kN$$

4. Sums — ΣFₓ = 11.49 kN, ΣF_y = 87.18 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(11.49^{2} + 87.18^{2}) = 87.93 kN at 82.5^{\circ}$$

Why the other options are there

• 152 kN (magnitudes added)
• 11.49 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 26 kN at 49° and 46 kN at 163° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$26cos49^{\circ} = 17.06 kN, 26sin49^{\circ} = 19.62 kN$$

3. F₂ components

   $$46cos163^{\circ} = -43.99 kN, 46sin163^{\circ} = 13.45 kN$$

4. Sums — ΣFₓ = -26.93 kN, ΣF_y = 33.07 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-26.93^{2} + 33.07^{2}) = 42.65 kN at 129.2^{\circ}$$

Why the other options are there

• 72 kN (magnitudes added)
• 26.93 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 47 kN at 53° and 53 kN at 120° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$47cos53^{\circ} = 28.29 kN, 47sin53^{\circ} = 37.54 kN$$

3. F₂ components

   $$53cos120^{\circ} = -26.50 kN, 53sin120^{\circ} = 45.90 kN$$

4. Sums — ΣFₓ = 1.79 kN, ΣF_y = 83.44 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(1.79^{2} + 83.44^{2}) = 83.45 kN at 88.8^{\circ}$$

Why the other options are there

• 100 kN (magnitudes added)
• 1.79 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 71 kN at 50° and 40 kN at 118° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$71cos50^{\circ} = 45.64 kN, 71sin50^{\circ} = 54.39 kN$$

3. F₂ components

   $$40cos118^{\circ} = -18.78 kN, 40sin118^{\circ} = 35.32 kN$$

4. Sums — ΣFₓ = 26.86 kN, ΣF_y = 89.71 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(26.86^{2} + 89.71^{2}) = 93.64 kN at 73.3^{\circ}$$

Why the other options are there

• 111 kN (magnitudes added)
• 26.86 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 63 kN at 71° and 69 kN at 122° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$63cos71^{\circ} = 20.51 kN, 63sin71^{\circ} = 59.57 kN$$

3. F₂ components

   $$69cos122^{\circ} = -36.56 kN, 69sin122^{\circ} = 58.52 kN$$

4. Sums — ΣFₓ = -16.05 kN, ΣF_y = 118.1 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-16.05^{2} + 118.1^{2}) = 119.2 kN at 97.7^{\circ}$$

Why the other options are there

• 132 kN (magnitudes added)
• 16.05 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 71 kN at 20° and 68 kN at 141° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$71cos20^{\circ} = 66.72 kN, 71sin20^{\circ} = 24.28 kN$$

3. F₂ components

   $$68cos141^{\circ} = -52.85 kN, 68sin141^{\circ} = 42.79 kN$$

4. Sums — ΣFₓ = 13.87 kN, ΣF_y = 67.08 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(13.87^{2} + 67.08^{2}) = 68.50 kN at 78.3^{\circ}$$

Why the other options are there

• 139 kN (magnitudes added)
• 13.87 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 43 kN at 17° and 90 kN at 135° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$43cos17^{\circ} = 41.12 kN, 43sin17^{\circ} = 12.57 kN$$

3. F₂ components

   $$90cos135^{\circ} = -63.64 kN, 90sin135^{\circ} = 63.64 kN$$

4. Sums — ΣFₓ = -22.52 kN, ΣF_y = 76.21 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-22.52^{2} + 76.21^{2}) = 79.47 kN at 106.5^{\circ}$$

Why the other options are there

• 133 kN (magnitudes added)
• 22.52 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 43 kN at 64° and 83 kN at 123° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$43cos64^{\circ} = 18.85 kN, 43sin64^{\circ} = 38.65 kN$$

3. F₂ components

   $$83cos123^{\circ} = -45.21 kN, 83sin123^{\circ} = 69.61 kN$$

4. Sums — ΣFₓ = -26.36 kN, ΣF_y = 108.3 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-26.36^{2} + 108.3^{2}) = 111.4 kN at 103.7^{\circ}$$

Why the other options are there

• 126 kN (magnitudes added)
• 26.36 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force

Two forces act at a gusset plate: 20 kN at 70° and 21 kN at 135° from the positive x-axis. Resolve each force into components and determine the magnitude and direction of the resultant force.

Start with the thinking

• A force system is resolved by summing x- and y-components separately.
• The resultant of concurrent forces is the vector sum, never the arithmetic sum of magnitudes.

Step-by-step solution

1. Formula

   $$Fₓ = F \cos \theta, F_y = F \sin \theta$$

2. F₁ components

   $$20cos70^{\circ} = 6.84 kN, 20sin70^{\circ} = 18.79 kN$$

3. F₂ components

   $$21cos135^{\circ} = -14.85 kN, 21sin135^{\circ} = 14.85 kN$$

4. Sums — ΣFₓ = -8.01 kN, ΣF_y = 33.64 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-8.01^{2} + 33.64^{2}) = 34.58 kN at 103.4^{\circ}$$

Why the other options are there

• 41 kN (magnitudes added)
• 8.01 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Force