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Step-by-Step Solution
Step 1: Identify the Given Quantities
Two forces have magnitudes Fâ and Fâ. We know:
The sum of their magnitudes is 18Â N, i.e.,
$F_1 + F_2 = 18$ .
The resultant R of these two forces has magnitude 12Â N and is perpendicular to the smaller force Fâ.
We are required to find the magnitudes of these two forces Fâ and Fâ.
Step 2: Express the Perpendicularity Condition
When the resultant R is perpendicular to the smaller force Fâ, it implies that:
$F_1^2 = F_2^2 + R^2$
(This comes from the Pythagorean theorem applied to the right triangle formed by Fâ, R, and Fâ.)
Step 3: Use Algebraic Manipulation
From the Pythagorean relation:
$F_1^2 - F_2^2 = R^2$
Substitute $R = 12$ :
$F_1^2 - F_2^2 = 12^2 = 144.$
Step 4: Factorize Using the Sum and Difference of Squares
The left-hand side $F_1^2 - F_2^2$ can be factorized as $(F_1 + F_2)(F_1 - F_2)$, so:
$(F_1 + F_2)(F_1 - F_2) = 144.$
Given $F_1 + F_2 = 18$ , substitute this into the equation:
$18 \times (F_1 - F_2) = 144.$
Thus,
$F_1 - F_2 = \frac{144}{18} = 8.$
Step 5: Solve the System of Equations
We now have two simultaneous equations:
$F_1 + F_2 = 18\,.$
$F_1 - F_2 = 8\,.$
Add these two equations:
$(F_1 + F_2) + (F_1 - F_2) = 18 + 8 \implies 2F_1 = 26 \implies F_1 = 13.$
Then substitute $F_1 = 13$ back into $F_1 + F_2 = 18$:
$13 + F_2 = 18 \implies F_2 = 5.$
Step 6: State the Final Answer
The magnitudes of the two forces are therefore:
$F_1 = 13\,\text{N}$ and $F_2 = 5\,\text{N}.$
