the greatest and least resultant of two forces are respectively 17 KN and 5 KN. determine the angle between two force when their resultant is root 144 KN.
Try to doodle a triangle of forces and find that the greatest and least resultants occur when the two forces (x,y) are colinear. When they are additive, we have x+y=17 kN, and when they oppose each other, we have x-y=5 (if x>y).
Thus solution of the system of equations yields
x=(17+5)/2=11
y=(17-5)/2=6
Going back to the triangle of forces for the resultant (z=√(144)=12), we have the relationship among x, y and z defined by the cosine rule, namely,
z²=x²+y²-2xy(cos(θ))
from which you can solve for θ=84° approx.
To find the angle between the two forces when their resultant is √144 KN, we need to use vector addition. The magnitude of the resultant vector R can be calculated using the Pythagorean theorem:
R = √(F1^2 + F2^2)
where F1 and F2 are the magnitudes of the two forces.
Given that the magnitudes of the greatest and least resultant forces are 17 KN and 5 KN, respectively, we have two equations:
17 = √(F1^2 + F2^2) (1)
5 = √(F1^2 + F2^2) (2)
From equation (2), we can square both sides to eliminate the square root:
25 = F1^2 + F2^2 (3)
From equation (1), we can square both sides as well:
289 = F1^2 + F2^2 (4)
Now we have a system of equations (3) and (4) for F1^2 and F2^2. Subtracting equation (3) from equation (4), we get:
289 - 25 = F1^2 + F2^2 - (F1^2 + F2^2)
264 = 0
This equation is not possible because 264 ≠ 0. Therefore, there are no values for F1 and F2 that satisfy both equations (1) and (2) simultaneously. As a result, we cannot determine the angle between the two forces when their resultant is √144 KN with the given information.
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