Two forces act at a gusset plate: 20 kN at 43° and 65 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

   $$20cos43^{\circ} = 14.63 kN, 20sin43^{\circ} = 13.64 kN$$

3. F₂ components

   $$65cos130^{\circ} = -41.78 kN, 65sin130^{\circ} = 49.79 kN$$

4. Sums — ΣFₓ = -27.15 kN, ΣF_y = 63.43 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-27.15^{2} + 63.43^{2}) = 69.00 kN at 113.2^{\circ}$$

Why the other options are there

• 85 kN (magnitudes added)
• 27.15 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 36 kN at 20° and 90 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

   $$36cos20^{\circ} = 33.83 kN, 36sin20^{\circ} = 12.31 kN$$

3. F₂ components

   $$90cos139^{\circ} = -67.92 kN, 90sin139^{\circ} = 59.05 kN$$

4. Sums — ΣFₓ = -34.09 kN, ΣF_y = 71.36 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-34.09^{2} + 71.36^{2}) = 79.08 kN at 115.5^{\circ}$$

Why the other options are there

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

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 80 kN at 37° and 58 kN at 126° 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

   $$80cos37^{\circ} = 63.89 kN, 80sin37^{\circ} = 48.15 kN$$

3. F₂ components

   $$58cos126^{\circ} = -34.09 kN, 58sin126^{\circ} = 46.92 kN$$

4. Sums — ΣFₓ = 29.80 kN, ΣF_y = 95.07 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(29.80^{2} + 95.07^{2}) = 99.63 kN at 72.6^{\circ}$$

Why the other options are there

• 138 kN (magnitudes added)
• 29.80 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 31 kN at 34° and 68 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

   $$31cos34^{\circ} = 25.70 kN, 31sin34^{\circ} = 17.33 kN$$

3. F₂ components

   $$68cos120^{\circ} = -34.00 kN, 68sin120^{\circ} = 58.89 kN$$

4. Sums — ΣFₓ = -8.30 kN, ΣF_y = 76.22 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-8.30^{2} + 76.22^{2}) = 76.68 kN at 96.2^{\circ}$$

Why the other options are there

• 99 kN (magnitudes added)
• 8.30 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 64 kN at 45° and 48 kN at 147° 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

   $$64cos45^{\circ} = 45.25 kN, 64sin45^{\circ} = 45.25 kN$$

3. F₂ components

   $$48cos147^{\circ} = -40.26 kN, 48sin147^{\circ} = 26.14 kN$$

4. Sums — ΣFₓ = 5.00 kN, ΣF_y = 71.40 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(5.00^{2} + 71.40^{2}) = 71.57 kN at 86.0^{\circ}$$

Why the other options are there

• 112 kN (magnitudes added)
• 5.00 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 87 kN at 57° and 74 kN at 143° 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

   $$87cos57^{\circ} = 47.38 kN, 87sin57^{\circ} = 72.96 kN$$

3. F₂ components

   $$74cos143^{\circ} = -59.10 kN, 74sin143^{\circ} = 44.53 kN$$

4. Sums — ΣFₓ = -11.72 kN, ΣF_y = 117.5 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-11.72^{2} + 117.5^{2}) = 118.1 kN at 95.7^{\circ}$$

Why the other options are there

• 161 kN (magnitudes added)
• 11.72 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 68 kN at 62° and 33 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

   $$68cos62^{\circ} = 31.92 kN, 68sin62^{\circ} = 60.04 kN$$

3. F₂ components

   $$33cos141^{\circ} = -25.65 kN, 33sin141^{\circ} = 20.77 kN$$

4. Sums — ΣFₓ = 6.28 kN, ΣF_y = 80.81 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(6.28^{2} + 80.81^{2}) = 81.05 kN at 85.6^{\circ}$$

Why the other options are there

• 101 kN (magnitudes added)
• 6.28 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 58 kN at 48° and 80 kN at 127° 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

   $$58cos48^{\circ} = 38.81 kN, 58sin48^{\circ} = 43.10 kN$$

3. F₂ components

   $$80cos127^{\circ} = -48.15 kN, 80sin127^{\circ} = 63.89 kN$$

4. Sums — ΣFₓ = -9.34 kN, ΣF_y = 107.0 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-9.34^{2} + 107.0^{2}) = 107.4 kN at 95.0^{\circ}$$

Why the other options are there

• 138 kN (magnitudes added)
• 9.34 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 68 kN at 22° and 33 kN at 165° 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

   $$68cos22^{\circ} = 63.05 kN, 68sin22^{\circ} = 25.47 kN$$

3. F₂ components

   $$33cos165^{\circ} = -31.88 kN, 33sin165^{\circ} = 8.54 kN$$

4. Sums — ΣFₓ = 31.17 kN, ΣF_y = 34.01 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(31.17^{2} + 34.01^{2}) = 46.14 kN at 47.5^{\circ}$$

Why the other options are there

• 101 kN (magnitudes added)
• 31.17 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force

Two forces act at a gusset plate: 83 kN at 50° and 89 kN at 154° 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

   $$83cos50^{\circ} = 53.35 kN, 83sin50^{\circ} = 63.58 kN$$

3. F₂ components

   $$89cos154^{\circ} = -79.99 kN, 89sin154^{\circ} = 39.02 kN$$

4. Sums — ΣFₓ = -26.64 kN, ΣF_y = 102.6 kN

5. Formula — R = √(ΣFₓ² + ΣF_y²)

6. Substituting

   $$R = \sqrt(-26.64^{2} + 102.6^{2}) = 106.0 kN at 104.6^{\circ}$$

Why the other options are there

• 172 kN (magnitudes added)
• 26.64 kN (y-component dropped)

Reference: FE Reference Handbook — Statics → Resolution of a Force