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The sum of the magnitudes of two forces acting at point is 18 and the magnitude of their resultant is 12. if the resultant is at 90° with force of smaller magnitude , what are the magnitudes of forces ?
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The sum of the magnitudes of two forces acting at point is 18 and the ...
Given Information
- The sum of the magnitudes of two forces (F1 + F2) = 18
- The magnitude of the resultant force (R) = 12
- The angle between the resultant and the smaller force (F1) = 90°
Understanding the Forces
When two forces act at a point, their resultant can be calculated using the Pythagorean theorem if they are at right angles. In this case:
- F1^2 + F2^2 = R^2
Since R = 12, we can substitute to find:
- F1^2 + F2^2 = 12^2
- F1^2 + F2^2 = 144
Setting Up the Equations
Given that F1 + F2 = 18, we can express F2 in terms of F1:
- F2 = 18 - F1
Substituting this into the equation for the resultant gives:
- F1^2 + (18 - F1)^2 = 144
Expanding the Equation
Now, we expand (18 - F1)^2:
- F1^2 + (324 - 36F1 + F1^2) = 144
- 2F1^2 - 36F1 + 324 = 144
Simplifying yields:
- 2F1^2 - 36F1 + 180 = 0
Solving the Quadratic Equation
Dividing through by 2:
- F1^2 - 18F1 + 90 = 0
Using the quadratic formula (F1 = [18 ± sqrt(18^2 - 4*90)]/2):
- F1 = 15 or 6
Thus,
- If F1 = 15, then F2 = 3.
- If F1 = 6, then F2 = 12.
Final Magnitudes
- The magnitudes of the forces are:
- F1 = 6, F2 = 12 (Smaller force)
- F1 = 15, F2 = 3 (Larger force)
Therefore, the solution provides the magnitudes of the two forces acting at the point, with one force being smaller than the other.
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The sum of the magnitudes of two forces acting at point is 18 and the magnitude of their resultant is 12. if the resultant is at 90° with force of smaller magnitude , what are the magnitudes of forces ?
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