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Two forces are such that the sum of their magnitudes is 18 N and their resultant which has magnitude 12 N, is perpendicular to the smaller force. Then the magnitudes of the forces are
- a)12 N, 6 N
- b)13 N, 5 N
- c)10 N, 8 N
- d)16 N, 2 N
Correct answer is option 'B'. Can you explain this answer?
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Two forces are such that the sum of their magnitudes is 18 N and their...
Solving Eqs. (i), (ii) and (iii), A = 5N, B = 13N
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Two forces are such that the sum of their magnitudes is 18 N and their...
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Two forces are such that the sum of their magnitudes is 18 N and their...
Given information:
- The sum of the magnitudes of two forces is 18 N.
- The resultant force has a magnitude of 12 N.
- The resultant force is perpendicular to the smaller force.
Let's assume the magnitudes of the two forces are F1 and F2, with F1 being the smaller force. According to the given information, we can write the following equations:
Equation 1: F1 + F2 = 18 N
Equation 2: √(F1^2 + F2^2) = 12 N
To find the magnitudes of the forces, we can solve these two equations simultaneously.
Solving the equations:
From Equation 2, we can square both sides to eliminate the square root:
F1^2 + F2^2 = 144
Now, let's square Equation 1:
(F1 + F2)^2 = (18 N)^2
F1^2 + 2F1F2 + F2^2 = 324
Substituting the value of F1^2 + F2^2 from the previous equation into this equation:
144 + 2F1F2 = 324
2F1F2 = 324 - 144
2F1F2 = 180
F1F2 = 90
Since the resultant force is perpendicular to the smaller force, we can use the concept of vectors to determine that the dot product of the two forces is zero:
F1 * F2 * cosθ = 0
Since cosθ = 0 (angle between the forces is 90 degrees), we have:
F1 * F2 = 0
Substituting the value of F1F2 = 90 into this equation, we get:
90 = 0
This is not a valid equation, so our assumption that cosθ = 0 is incorrect. Therefore, cosθ ≠ 0, which implies that the angle between the forces is not 90 degrees.
Conclusion:
Based on the given information, it is not possible to determine the magnitudes of the forces. Therefore, none of the given options (A, B, C, D) are correct.
- The sum of the magnitudes of two forces is 18 N.
- The resultant force has a magnitude of 12 N.
- The resultant force is perpendicular to the smaller force.
Let's assume the magnitudes of the two forces are F1 and F2, with F1 being the smaller force. According to the given information, we can write the following equations:
Equation 1: F1 + F2 = 18 N
Equation 2: √(F1^2 + F2^2) = 12 N
To find the magnitudes of the forces, we can solve these two equations simultaneously.
Solving the equations:
From Equation 2, we can square both sides to eliminate the square root:
F1^2 + F2^2 = 144
Now, let's square Equation 1:
(F1 + F2)^2 = (18 N)^2
F1^2 + 2F1F2 + F2^2 = 324
Substituting the value of F1^2 + F2^2 from the previous equation into this equation:
144 + 2F1F2 = 324
2F1F2 = 324 - 144
2F1F2 = 180
F1F2 = 90
Since the resultant force is perpendicular to the smaller force, we can use the concept of vectors to determine that the dot product of the two forces is zero:
F1 * F2 * cosθ = 0
Since cosθ = 0 (angle between the forces is 90 degrees), we have:
F1 * F2 = 0
Substituting the value of F1F2 = 90 into this equation, we get:
90 = 0
This is not a valid equation, so our assumption that cosθ = 0 is incorrect. Therefore, cosθ ≠ 0, which implies that the angle between the forces is not 90 degrees.
Conclusion:
Based on the given information, it is not possible to determine the magnitudes of the forces. Therefore, none of the given options (A, B, C, D) are correct.
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