Two cables pulling on an eyebolt are concurrent forces summed to a resultant. Given force 1 (F_1) = 260.0 lbf; force 2 (F_2) = 102.0 lbf; angle between forces (theta) = 1.9200 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 260.0 lbf, force 2 (F_2) = 102.0 lbf, angle between forces (theta) = 1.9200 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos1.9200}$$

5. Step 5 — Evaluate:

   $$R = 244.7\ \text{lbf}$$

6. Step 6 — Check: returning R = 244.7 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 489.3 — kept a factor of two that cancels in the correct rearrangement.
• 122.3 — dropped that same factor in the other direction.
• 269.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A crane hook experiences concurrent forces from two sling legs. Given force 2 (F_2) = 206.0 lbf; angle between forces (theta) = 0.3200 rad; resultant force (R) = 381.0 lbf, determine the force 1 (F_1) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except F_1 is given, so isolate F_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for F_1:

   $$F_{1} = -F_2\cos\theta + \sqrt{R^2 - F_2^2\sin^2\theta}$$

3. Step 3 — List the givens: force 2 (F_2) = 206.0 lbf, angle between forces (theta) = 0.3200 rad, resultant force (R) = 381.0 lbf.

4. Step 4 — Substitute the given values:

   $$F_{1} = -F_2\cos0.3200 + \sqrt{381.0^2 - F_2^2\sin^20.3200}$$

5. Step 5 — Evaluate:

   $$F_{1} = 179.9\ \text{lbf}$$

6. Step 6 — Check: returning F_1 = 179.9 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 359.8 — kept a factor of two that cancels in the correct rearrangement.
• 89.9532 — dropped that same factor in the other direction.
• 197.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A truss joint has concurrent forces from two diagonal members meeting at a pin. Given force 1 (F_1) = 157.0 lbf; force 2 (F_2) = 265.0 lbf; angle between forces (theta) = 2.7400 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 157.0 lbf, force 2 (F_2) = 265.0 lbf, angle between forces (theta) = 2.7400 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos2.7400}$$

5. Step 5 — Evaluate:

   $$R = 135.2\ \text{lbf}$$

6. Step 6 — Check: returning R = 135.2 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 270.4 — kept a factor of two that cancels in the correct rearrangement.
• 67.6096 — dropped that same factor in the other direction.
• 148.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

Two cables pulling on an eyebolt are concurrent forces summed to a resultant. Given force 2 (F_2) = 64.0000 lbf; angle between forces (theta) = 1.8200 rad; resultant force (R) = 501.0 lbf, determine the force 1 (F_1) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except F_1 is given, so isolate F_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for F_1:

   $$F_{1} = -F_2\cos\theta + \sqrt{R^2 - F_2^2\sin^2\theta}$$

3. Step 3 — List the givens: force 2 (F_2) = 64.0000 lbf, angle between forces (theta) = 1.8200 rad, resultant force (R) = 501.0 lbf.

4. Step 4 — Substitute the given values:

   $$F_{1} = -F_2\cos1.8200 + \sqrt{501.0^2 - F_2^2\sin^21.8200}$$

5. Step 5 — Evaluate:

   $$F_{1} = 512.9\ \text{lbf}$$

6. Step 6 — Check: returning F_1 = 512.9 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,026 — kept a factor of two that cancels in the correct rearrangement.
• 256.5 — dropped that same factor in the other direction.
• 564.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A crane hook experiences concurrent forces from two sling legs. Given force 1 (F_1) = 125.0 lbf; force 2 (F_2) = 81.0000 lbf; angle between forces (theta) = 1.0400 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 125.0 lbf, force 2 (F_2) = 81.0000 lbf, angle between forces (theta) = 1.0400 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos1.0400}$$

5. Step 5 — Evaluate:

   $$R = 180.1\ \text{lbf}$$

6. Step 6 — Check: returning R = 180.1 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 360.2 — kept a factor of two that cancels in the correct rearrangement.
• 90.0513 — dropped that same factor in the other direction.
• 198.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

Two cables pulling on an eyebolt are concurrent forces summed to a resultant. Given force 1 (F_1) = 182.0 lbf; force 2 (F_2) = 208.0 lbf; angle between forces (theta) = 1.3200 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 182.0 lbf, force 2 (F_2) = 208.0 lbf, angle between forces (theta) = 1.3200 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos1.3200}$$

5. Step 5 — Evaluate:

   $$R = 308.5\ \text{lbf}$$

6. Step 6 — Check: returning R = 308.5 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 617.0 — kept a factor of two that cancels in the correct rearrangement.
• 154.3 — dropped that same factor in the other direction.
• 339.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A crane hook experiences concurrent forces from two sling legs. Given force 2 (F_2) = 41.0000 lbf; angle between forces (theta) = 2.4600 rad; resultant force (R) = 56.0000 lbf, determine the force 1 (F_1) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except F_1 is given, so isolate F_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for F_1:

   $$F_{1} = -F_2\cos\theta + \sqrt{R^2 - F_2^2\sin^2\theta}$$

3. Step 3 — List the givens: force 2 (F_2) = 41.0000 lbf, angle between forces (theta) = 2.4600 rad, resultant force (R) = 56.0000 lbf.

4. Step 4 — Substitute the given values:

   $$F_{1} = -F_2\cos2.4600 + \sqrt{56.0000^2 - F_2^2\sin^22.4600}$$

5. Step 5 — Evaluate:

   $$F_{1} = 81.5259\ \text{lbf}$$

6. Step 6 — Check: returning F_1 = 81.5259 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 163.1 — kept a factor of two that cancels in the correct rearrangement.
• 40.7629 — dropped that same factor in the other direction.
• 89.6784 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A truss joint has concurrent forces from two diagonal members meeting at a pin. Given force 1 (F_1) = 212.0 lbf; force 2 (F_2) = 87.0000 lbf; angle between forces (theta) = 1.0000 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 212.0 lbf, force 2 (F_2) = 87.0000 lbf, angle between forces (theta) = 1.0000 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos1.0000}$$

5. Step 5 — Evaluate:

   $$R = 269.2\ \text{lbf}$$

6. Step 6 — Check: returning R = 269.2 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 538.3 — kept a factor of two that cancels in the correct rearrangement.
• 134.6 — dropped that same factor in the other direction.
• 296.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A crane hook experiences concurrent forces from two sling legs. Given force 1 (F_1) = 31.0000 lbf; force 2 (F_2) = 55.0000 lbf; angle between forces (theta) = 1.0000 rad, determine the resultant force (R) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

3. Step 3 — List the givens: force 1 (F_1) = 31.0000 lbf, force 2 (F_2) = 55.0000 lbf, angle between forces (theta) = 1.0000 rad.

4. Step 4 — Substitute the given values:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos1.0000}$$

5. Step 5 — Evaluate:

   $$R = 76.3442\ \text{lbf}$$

6. Step 6 — Check: returning R = 76.3442 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 152.7 — kept a factor of two that cancels in the correct rearrangement.
• 38.1721 — dropped that same factor in the other direction.
• 83.9786 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces

A truss joint has concurrent forces from two diagonal members meeting at a pin. Given force 2 (F_2) = 61.0000 lbf; angle between forces (theta) = 1.4800 rad; resultant force (R) = 500.0 lbf, determine the force 1 (F_1) in lbf.

Start with the thinking

• The governing relation printed in this handbook section is Resultant of two concurrent forces.
• Everything except F_1 is given, so isolate F_1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Two concurrent forces acting at a point are combined into a single resultant using the law of cosines for concurrent forces.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

2. Step 2 — Rearrange symbolically for F_1:

   $$F_{1} = -F_2\cos\theta + \sqrt{R^2 - F_2^2\sin^2\theta}$$

3. Step 3 — List the givens: force 2 (F_2) = 61.0000 lbf, angle between forces (theta) = 1.4800 rad, resultant force (R) = 500.0 lbf.

4. Step 4 — Substitute the given values:

   $$F_{1} = -F_2\cos1.4800 + \sqrt{500.0^2 - F_2^2\sin^21.4800}$$

5. Step 5 — Evaluate:

   $$F_{1} = 490.8\ \text{lbf}$$

6. Step 6 — Check: returning F_1 = 490.8 lbf to

   $$R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos\theta}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 981.5 — kept a factor of two that cancels in the correct rearrangement.
• 245.4 — dropped that same factor in the other direction.
• 539.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Concurrent Forces