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A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?
- a)80i + 40j – 80k N
- b)80i – 40j – 80k N
- c)151i + 40j – 80k N
- d)151i+ 40j – 151k N
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A force vector is along 4i – 4k direction and has a magnitude 10...
Just make the unit vectors of the given directions of the vectors and then multiply them with their respective magnitudes, after than you will get two vectors in the Cartesian form. Just add them.
Most Upvoted Answer
A force vector is along 4i – 4k direction and has a magnitude 10...
Multiply magnitude of each vector with its unit direction and add.
(for unit vector,divide direction vector by magnitude.)
(for unit vector,divide direction vector by magnitude.)
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Community Answer
A force vector is along 4i – 4k direction and has a magnitude 10...
Given Force Vectors
- First Force Vector (F1):
- Direction: \(4\hat{i} - 4\hat{k}\)
- Magnitude: \(100 N\)
- Second Force Vector (F2):
- Direction: \(4\hat{i} + 2\hat{j} - 4\hat{k}\)
- Magnitude: \(120 N\)
Calculating Unit Vectors
1. For F1:
- Direction vector: \( \vec{d_1} = 4\hat{i} - 4\hat{k} \)
- Magnitude of direction: \( |\vec{d_1}| = \sqrt{4^2 + 0^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2} \)
- Unit vector: \( \hat{u_1} = \frac{1}{4\sqrt{2}}(4\hat{i} - 4\hat{k}) = \hat{i} - \hat{k} \)
- Force vector \( \vec{F_1} = 100(\hat{i} - \hat{k}) = 100\hat{i} - 100\hat{k} \)
2. For F2:
- Direction vector: \( \vec{d_2} = 4\hat{i} + 2\hat{j} - 4\hat{k} \)
- Magnitude of direction: \( |\vec{d_2}| = \sqrt{4^2 + 2^2 + (-4)^2} = \sqrt{36} = 6 \)
- Unit vector: \( \hat{u_2} = \frac{1}{6}(4\hat{i} + 2\hat{j} - 4\hat{k}) = \frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k} \)
- Force vector \( \vec{F_2} = 120\hat{u_2} = 120\left(\frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}\right) = 80\hat{i} + 40\hat{j} - 80\hat{k} \)
Resultant Force Vector Calculation
- Adding Force Vectors:
- \( \vec{F_R} = \vec{F_1} + \vec{F_2} = (100\hat{i} - 100\hat{k}) + (80\hat{i} + 40\hat{j} - 80\hat{k}) \)
- Combine Components:
- \( \vec{F_R} = (100 + 80)\hat{i} + 40\hat{j} + (-100 - 80)\hat{k} \)
- \( \vec{F_R} = 180\hat{i} + 40\hat{j} - 180\hat{k} \)
Conclusion
The correct
- First Force Vector (F1):
- Direction: \(4\hat{i} - 4\hat{k}\)
- Magnitude: \(100 N\)
- Second Force Vector (F2):
- Direction: \(4\hat{i} + 2\hat{j} - 4\hat{k}\)
- Magnitude: \(120 N\)
Calculating Unit Vectors
1. For F1:
- Direction vector: \( \vec{d_1} = 4\hat{i} - 4\hat{k} \)
- Magnitude of direction: \( |\vec{d_1}| = \sqrt{4^2 + 0^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2} \)
- Unit vector: \( \hat{u_1} = \frac{1}{4\sqrt{2}}(4\hat{i} - 4\hat{k}) = \hat{i} - \hat{k} \)
- Force vector \( \vec{F_1} = 100(\hat{i} - \hat{k}) = 100\hat{i} - 100\hat{k} \)
2. For F2:
- Direction vector: \( \vec{d_2} = 4\hat{i} + 2\hat{j} - 4\hat{k} \)
- Magnitude of direction: \( |\vec{d_2}| = \sqrt{4^2 + 2^2 + (-4)^2} = \sqrt{36} = 6 \)
- Unit vector: \( \hat{u_2} = \frac{1}{6}(4\hat{i} + 2\hat{j} - 4\hat{k}) = \frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k} \)
- Force vector \( \vec{F_2} = 120\hat{u_2} = 120\left(\frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}\right) = 80\hat{i} + 40\hat{j} - 80\hat{k} \)
Resultant Force Vector Calculation
- Adding Force Vectors:
- \( \vec{F_R} = \vec{F_1} + \vec{F_2} = (100\hat{i} - 100\hat{k}) + (80\hat{i} + 40\hat{j} - 80\hat{k}) \)
- Combine Components:
- \( \vec{F_R} = (100 + 80)\hat{i} + 40\hat{j} + (-100 - 80)\hat{k} \)
- \( \vec{F_R} = 180\hat{i} + 40\hat{j} - 180\hat{k} \)
Conclusion
The correct
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A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?a)80i + 40j – 80k Nb)80i – 40j – 80k Nc)151i + 40j – 80k Nd)151i+ 40j – 151k NCorrect answer is option 'D'. Can you explain this answer?
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A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?a)80i + 40j – 80k Nb)80i – 40j – 80k Nc)151i + 40j – 80k Nd)151i+ 40j – 151k NCorrect answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?a)80i + 40j – 80k Nb)80i – 40j – 80k Nc)151i + 40j – 80k Nd)151i+ 40j – 151k NCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?a)80i + 40j – 80k Nb)80i – 40j – 80k Nc)151i + 40j – 80k Nd)151i+ 40j – 151k NCorrect answer is option 'D'. Can you explain this answer?.
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