08 Resolution of Force Problem Set
RESOLUTION OF A FORCE | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.
PROBLEM 1: A hurricane wind blowing ๐๐ก 300 ๐๐๐๐๐ ๐๐๐ โ๐๐ข๐ is acting on a building in Sunrise, FL is defined by the force vector ๐น = 3.5๐ โ 1.5๐ + 2.0๐. What is the angle that the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 66.4ยฐ C. 69.6ยฐ D. 110ยฐ
SOLUTION 1: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing.
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We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
3.5
B
+ โ1.5
B
+ 2.0 B = 4.3
We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฆ โ ๐๐ฅ๐๐ usually represents the vertical component of the force:
๐นE = cos ๐E ๐๐ ๐๐๐ ๐E =
๐นE ๐น
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As we are solving for the angles the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the y-axis to solve for the angle:
๐ = cos NO
๐นE ๐น
Plugging in the given values for the y-component, and the calculated magnitude of the force, we find the angle between the force and the y-axis is:
๐ = cos NO โ
1.5 = 110.4ยฐ 4.3
Therefore, the correct answer choice is D. ๐๐๐ยฐ
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PROBLEM 2: A transmission tower is supported by a guy wire that anchored to the ground at point A and attached to the top of the tower at point B. If the cable has a tension of 2,500, what is the resultant of the tensile force closest to:
A. 235,850 B. 243,280 C. 246,800 D. 268,000
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SOLUTION 2: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The line of action of the force acting on the bolt passes through points A and B, and the force is directed from A to B. The components of the vector AB are represented by the distances: ๐D = โ40 ๐ ๐E = +80 ๐ ๐F = +30 ๐ We can write the force for the cable in vector form as: ๐น = 2,500 โ40๐ + 80๐ + 30๐ = โ100,000๐ + 200,000๐ + 75,000๐ We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
โ100,000
B
+ 200,000
B
+ 75,000 B = 235,849.53
Therefore, the correct answer choice is A. ๐๐๐, ๐๐๐
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PROBLEM 3: A force is represented by the force vector ๐น = โ1060๐ + 2120๐ + 795๐, what is the angle that the force makes with the positive ๐ฅ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 32.0ยฐ C. 71.5ยฐ D. 115.1ยฐ
SOLUTION 3: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
1060
B
+ 2120
B
+ 795 B = 2500
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We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฅ โ ๐๐ฅ๐๐ usually represents the horizontal component of the force:
๐นD = cos ๐D ๐๐ ๐๐๐ ๐D =
๐นD ๐น
As we are solving for the angles the force makes with the positive ๐ฅ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the x-axis to solve for the angle:
๐ = cos NO
๐นD ๐น
Plugging in the given values for the x-component, and the calculated magnitude of the force, we find the angle between the force and the ๐ฅ โ ๐๐ฅ๐๐ is:
๐ = cos NO โ
1060 = 115.1ยฐ 2500
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Therefore, the correct answer choice is D. ๐๐๐. ๐ยฐ PROBLEM 4: A force is represented by the force vector ๐น = โ1060๐ + 2120๐ + 795๐, what is the angle that the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 32.0ยฐ C. 71.5 D. 115.1ยฐ
SOLUTION 4: The FORMULA FOR THE RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
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Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
1060
B
+ 2120
B
+ 795 B = 2500
We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฆ โ ๐๐ฅ๐๐ usually represents the vertical component of the force:
๐นE = cos ๐E ๐๐ ๐๐๐ ๐E =
๐นE ๐น
As we are solving for the angles the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the ๐ฆ โ ๐๐ฅ๐๐ to solve for the angle:
๐ = cos NO
๐นE ๐น
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Plugging in the given values for the y-component, and the calculated magnitude of the force, we find the angle between the force and the ๐ฆ โ ๐๐ฅ๐๐ is:
๐ = cos NO
2120 = 32.0ยฐ 2500
Therefore, the correct answer choice is B. ๐๐. ๐ยฐ
PROBLEM 5: A particle is in equilibrium is said to be in three-dimensional static equilibrium with 10 forces acting on it. How many equations of equilibrium can be written for the particle? A. 2 B. 3 C. 4 D. ๐๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ฃ๐
SOLUTION 5: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z).
Therefore, the correct answer choice is B. ๐
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PROBLEM 6: A particle is in equilibrium is said to be in three-dimensional static equilibrium with no external forces acting on it. Which of the following equations would represent the static equilibrium of the particle? A. โ๐นD ๐ + โ๐นE ๐ + โ๐นF ๐ = 0 B. โ๐น = 0 C. โ๐นD = โ๐นE = โ๐นF = 0 D. ๐ด๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ฃ๐
SOLUTION 6: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z). As we are told that the particle is in static equilibrium, we know that the sum of forces about any plane or axis will be equal to zero. Looking at the answer choices, we see that all three are different forms of expressing the equations of equilibrium of 3-D particle in static equilibrium.
Therefore, the correct answer choice is D. ๐จ๐๐ ๐๐ ๐๐๐ ๐๐๐๐๐
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PROBLEM 7: When working with three-dimensional problems and you are not given the direction or the magnitude of a force, how many unknowns do you have corresponding to that force? A. ๐๐๐ B. ๐๐ค๐ C. ๐โ๐๐๐ D. ๐น๐๐ข๐
SOLUTION 7: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z). As we have three dimensions and do not have the magnitude of the force, we will have three components of the force that we will need to solve for.
Therefore, the correct answer choice is C. ๐ป๐๐๐๐
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PROBLEM 1: A hurricane wind blowing ๐๐ก 300 ๐๐๐๐๐ ๐๐๐ โ๐๐ข๐ is acting on a building in Sunrise, FL is defined by the force vector ๐น = 3.5๐ โ 1.5๐ + 2.0๐. What is the angle that the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 66.4ยฐ C. 69.6ยฐ D. 110ยฐ
SOLUTION 1: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing.
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We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
3.5
B
+ โ1.5
B
+ 2.0 B = 4.3
We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฆ โ ๐๐ฅ๐๐ usually represents the vertical component of the force:
๐นE = cos ๐E ๐๐ ๐๐๐ ๐E =
๐นE ๐น
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As we are solving for the angles the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the y-axis to solve for the angle:
๐ = cos NO
๐นE ๐น
Plugging in the given values for the y-component, and the calculated magnitude of the force, we find the angle between the force and the y-axis is:
๐ = cos NO โ
1.5 = 110.4ยฐ 4.3
Therefore, the correct answer choice is D. ๐๐๐ยฐ
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PROBLEM 2: A transmission tower is supported by a guy wire that anchored to the ground at point A and attached to the top of the tower at point B. If the cable has a tension of 2,500, what is the resultant of the tensile force closest to:
A. 235,850 B. 243,280 C. 246,800 D. 268,000
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SOLUTION 2: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The line of action of the force acting on the bolt passes through points A and B, and the force is directed from A to B. The components of the vector AB are represented by the distances: ๐D = โ40 ๐ ๐E = +80 ๐ ๐F = +30 ๐ We can write the force for the cable in vector form as: ๐น = 2,500 โ40๐ + 80๐ + 30๐ = โ100,000๐ + 200,000๐ + 75,000๐ We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
โ100,000
B
+ 200,000
B
+ 75,000 B = 235,849.53
Therefore, the correct answer choice is A. ๐๐๐, ๐๐๐
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PROBLEM 3: A force is represented by the force vector ๐น = โ1060๐ + 2120๐ + 795๐, what is the angle that the force makes with the positive ๐ฅ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 32.0ยฐ C. 71.5ยฐ D. 115.1ยฐ
SOLUTION 3: The formula for the RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
1060
B
+ 2120
B
+ 795 B = 2500
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We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฅ โ ๐๐ฅ๐๐ usually represents the horizontal component of the force:
๐นD = cos ๐D ๐๐ ๐๐๐ ๐D =
๐นD ๐น
As we are solving for the angles the force makes with the positive ๐ฅ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the x-axis to solve for the angle:
๐ = cos NO
๐นD ๐น
Plugging in the given values for the x-component, and the calculated magnitude of the force, we find the angle between the force and the ๐ฅ โ ๐๐ฅ๐๐ is:
๐ = cos NO โ
1060 = 115.1ยฐ 2500
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Therefore, the correct answer choice is D. ๐๐๐. ๐ยฐ PROBLEM 4: A force is represented by the force vector ๐น = โ1060๐ + 2120๐ + 795๐, what is the angle that the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ ? A. 20.4ยฐ B. 32.0ยฐ C. 71.5 D. 115.1ยฐ
SOLUTION 4: The FORMULA FOR THE RESULTANT OF A THREE-DIMENSIONAL FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. We can calculate the resultant of a three-dimensional force by calculating the magnitude of each component.
๐ =
๐ฅB + ๐ฆB + ๐งB =
๐นDB + ๐นEB + ๐นFB
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Plugging in the value for each component, we calculate the resultant of the force as:
๐ =
1060
B
+ 2120
B
+ 795 B = 2500
We can rewrite the formula for the magnitude of each component by substituting in the formula for the resultant. Therefore, we can solve for the component of each force by using the formula for the resultant of a three-dimensional force:
๐นD =
๐ฅ ๐ฆ ๐ง ๐น; ๐นE = ๐น; ๐นF = ๐น ๐ ๐ ๐
The formulas for the RESOLUTION OF A FORCE can be referenced under the topic of STATICS on page 67 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. The ๐ฆ โ ๐๐ฅ๐๐ usually represents the vertical component of the force:
๐นE = cos ๐E ๐๐ ๐๐๐ ๐E =
๐นE ๐น
As we are solving for the angles the force makes with the positive ๐ฆ โ ๐๐ฅ๐๐ , we can rewrite the direction cosine for the ๐ฆ โ ๐๐ฅ๐๐ to solve for the angle:
๐ = cos NO
๐นE ๐น
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Plugging in the given values for the y-component, and the calculated magnitude of the force, we find the angle between the force and the ๐ฆ โ ๐๐ฅ๐๐ is:
๐ = cos NO
2120 = 32.0ยฐ 2500
Therefore, the correct answer choice is B. ๐๐. ๐ยฐ
PROBLEM 5: A particle is in equilibrium is said to be in three-dimensional static equilibrium with 10 forces acting on it. How many equations of equilibrium can be written for the particle? A. 2 B. 3 C. 4 D. ๐๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ฃ๐
SOLUTION 5: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z).
Therefore, the correct answer choice is B. ๐
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PROBLEM 6: A particle is in equilibrium is said to be in three-dimensional static equilibrium with no external forces acting on it. Which of the following equations would represent the static equilibrium of the particle? A. โ๐นD ๐ + โ๐นE ๐ + โ๐นF ๐ = 0 B. โ๐น = 0 C. โ๐นD = โ๐นE = โ๐นF = 0 D. ๐ด๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ฃ๐
SOLUTION 6: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z). As we are told that the particle is in static equilibrium, we know that the sum of forces about any plane or axis will be equal to zero. Looking at the answer choices, we see that all three are different forms of expressing the equations of equilibrium of 3-D particle in static equilibrium.
Therefore, the correct answer choice is D. ๐จ๐๐ ๐๐ ๐๐๐ ๐๐๐๐๐
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PROBLEM 7: When working with three-dimensional problems and you are not given the direction or the magnitude of a force, how many unknowns do you have corresponding to that force? A. ๐๐๐ B. ๐๐ค๐ C. ๐โ๐๐๐ D. ๐น๐๐ข๐
SOLUTION 7: When working with problems in three-dimensions, we can only sum the forces around each axis that a force or forces is acting about. Therefore, as we have 3 dimensions or plans of equilibrium, the maximum amount of equilibrium equations we can write is three (x, y, and z). As we have three dimensions and do not have the magnitude of the force, we will have three components of the force that we will need to solve for.
Therefore, the correct answer choice is C. ๐ป๐๐๐๐
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