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The sum of two forces is 18 N and resultant whose direction is at right angles to the smaller force is 12 N. The magnitude of the two forces are [2002]
- a)13, 5
- b)12, 6
- c)14, 4
- d)11, 7
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The sum of two forces is 18 N and resultant whose direction is at righ...
Given P + Q = 18 .......(1)
P2 + Q2 + 2PQ cosα = 144 ......(2)
P2 + Q2 + 2PQ cosα = 144 ......(2)
⇒ P + Q cosα = 0 ......(3)
From (2) and (3),
Q2 – P2 = 144 ⇒ (Q – P) (Q + P) = 144
From (2) and (3),
Q2 – P2 = 144 ⇒ (Q – P) (Q + P) = 144
From (1), On solving, we get Q = 13, P = 5
Most Upvoted Answer
The sum of two forces is 18 N and resultant whose direction is at righ...
Given information:
- The sum of two forces is 18 N.
- The resultant force, which is at right angles to the smaller force, is 12 N.
To find:
The magnitude of the two forces.
Let's assume the magnitude of the smaller force is x N. Therefore, the magnitude of the larger force would be (18 - x) N, as the sum of the forces is 18 N.
Using Pythagoras' theorem, we can find the magnitude of the resultant force:
Resultant force^2 = smaller force^2 + larger force^2
(12 N)^2 = x^2 + (18 - x)^2
Simplifying the equation:
144 N^2 = x^2 + (18 - x)^2
144 N^2 = x^2 + 324 - 36x + x^2
288 N^2 = 2x^2 - 36x + 324
2x^2 - 36x + 324 - 288 = 0
2x^2 - 36x + 36 = 0
Dividing the equation by 2:
x^2 - 18x + 18 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
x = (-(-18) ± √((-18)^2 - 4(1)(18))) / (2(1))
x = (18 ± √(324 - 72)) / 2
x = (18 ± √252) / 2
x = (18 ± 2√63) / 2
x = 9 ± √63
Since the magnitude of a force cannot be negative, we take the positive value:
x = 9 + √63
Therefore, the magnitude of the smaller force is 9 + √63 N, and the magnitude of the larger force is 18 - (9 + √63) N.
Calculating the values:
Magnitude of the smaller force = 9 + √63 ≈ 13 N
Magnitude of the larger force = 18 - (9 + √63) ≈ 5 N
Therefore, the magnitude of the two forces is approximately 13 N and 5 N, which matches with option A.
- The sum of two forces is 18 N.
- The resultant force, which is at right angles to the smaller force, is 12 N.
To find:
The magnitude of the two forces.
Let's assume the magnitude of the smaller force is x N. Therefore, the magnitude of the larger force would be (18 - x) N, as the sum of the forces is 18 N.
Using Pythagoras' theorem, we can find the magnitude of the resultant force:
Resultant force^2 = smaller force^2 + larger force^2
(12 N)^2 = x^2 + (18 - x)^2
Simplifying the equation:
144 N^2 = x^2 + (18 - x)^2
144 N^2 = x^2 + 324 - 36x + x^2
288 N^2 = 2x^2 - 36x + 324
2x^2 - 36x + 324 - 288 = 0
2x^2 - 36x + 36 = 0
Dividing the equation by 2:
x^2 - 18x + 18 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
x = (-(-18) ± √((-18)^2 - 4(1)(18))) / (2(1))
x = (18 ± √(324 - 72)) / 2
x = (18 ± √252) / 2
x = (18 ± 2√63) / 2
x = 9 ± √63
Since the magnitude of a force cannot be negative, we take the positive value:
x = 9 + √63
Therefore, the magnitude of the smaller force is 9 + √63 N, and the magnitude of the larger force is 18 - (9 + √63) N.
Calculating the values:
Magnitude of the smaller force = 9 + √63 ≈ 13 N
Magnitude of the larger force = 18 - (9 + √63) ≈ 5 N
Therefore, the magnitude of the two forces is approximately 13 N and 5 N, which matches with option A.
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The sum of two forces is 18 N and resultant whose direction is at right angles to the smaller force is 12 N. The magnitude of the two forces are [2002]a)13, 5b)12, 6c)14, 4d)11, 7Correct answer is option 'A'. Can you explain this answer?
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