Find the megnitude of 2 forces.such that if they act at triangular,there is resultant √10N . If act at 60° the resultant is √13N ?
To find the magnitudes of the two forces, we can use the law of cosines.
Let's call the magnitudes of the two forces A and B. We'll use triangle ABC, where A and B are the sides of length A and B, and C is the side opposite the angle of 60°.
Using the law of cosines, we can find the magnitude of A:
A^2 = B^2 + C^2 - 2BC * cos(60°)
Substituting the given values:
A^2 = B^2 + (√13N)^2 - 2B * √13N * cos(60°) [1]
Similarly, we can find the magnitude of B:
B^2 = A^2 + C^2 - 2AC * cos(60°)
Substituting the given values:
B^2 = A^2 + (√10N)^2 - 2A * √10N * cos(60°) [2]
Now, we have a system of equations [1] and [2]. We can solve this system of equations to find the values of A and B.
Let's solve the system of equations:
From equation [1]:
A^2 = B^2 + 13 - 2B * √13N * 0.5
= B^2 + 13 - B * √13N [3]
From equation [2]:
B^2 = A^2 + 10 - 2A * √10N * 0.5
= A^2 + 10 - A * √10N [4]
Substituting equation [3] into equation [4]:
B^2 = (B^2 + 13 - B * √13N) + 10 - (B * √13N)
B^2 = B^2 + 10 + 13 - B * √13N - B * √13N
-20 = - 2B * √13N
Dividing both sides by -2√13N:
10 = B
Substituting the value of B into equation [3]:
A^2 = (10)^2 + 13 - 10 * √13N
A^2 = 100 + 13 - 10 * √13N
A^2 = 113 - 10 * √13N
A ≈ √(113 - 10√13N)
Therefore, the magnitudes of the two forces are approximately:
Magnitude of force A ≈ √(113 - 10√13N)
Magnitude of force B = 10