the greatest and the least resultant of twoforces acting on a particle are the 35KN and 5 KN respectively. if 25KN is the magnitude of the resultant for the given system of forces f1 and f2 .prove that the forces are acting at right angle
a+b = 35
a-b = 5
so
a = 20
b = 15
c^2 = a^2+b^2 - 2 a b cos C
625 = 400 + 225 - 2 a b cos C
so cos C = 0
C = 90 or 270 degrees
To prove that the two forces are acting at a right angle, we can make use of the Pythagorean theorem and vector addition.
Let's denote the magnitude of the two forces as F1 and F2, where F1 is the greater force and F2 is the smaller force.
According to the given information, F1 = 35 kN and F2 = 5 kN, and the magnitude of the resultant force is R = 25 kN.
Using the Pythagorean theorem, we know that for a right-angled triangle:
R^2 = F1^2 + F2^2
Substituting the given values, we have:
(25 kN)^2 = (35 kN)^2 + (5 kN)^2
625 kN^2 = 1225 kN^2 + 25 kN^2
Now, let's subtract (35 kN)^2 from both sides:
625 kN^2 - (35 kN)^2 = 25 kN^2
390 kN^2 = 25 kN^2
Since 390 kN^2 is not equal to 25 kN^2, we have reached a contradiction. Therefore, it is not possible for the forces to be acting at a right angle.
Hence, we cannot prove that the forces are acting at a right angle based on the given information and calculations.
To prove that the two forces are acting at a right angle, we need to use the concept of vector addition and the Pythagorean theorem.
Let's assume that F1 and F2 are the magnitudes of the two forces acting on a particle, and R is the magnitude of the resultant force. Given that the greatest resultant is 35 KN and the least resultant is 5 KN, we have:
F1 + F2 = 35 KN (Equation 1)
F1 - F2 = 5 KN (Equation 2)
We can solve these two equations to find the values of F1 and F2. Here's how:
1. Add Equation 1 and Equation 2:
(F1 + F2) + (F1 - F2) = 35 KN + 5 KN
2F1 = 40 KN
F1 = 20 KN
2. Substitute the value of F1 into Equation 1:
20 KN + F2 = 35 KN
F2 = 35 KN - 20 KN
F2 = 15 KN
Now we know that F1 = 20 KN and F2 = 15 KN.
To prove that the forces are acting at right angles, we can use the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the hypotenuse (R) is equal to the sum of the squares of the other two sides (F1 and F2).
R^2 = F1^2 + F2^2
(25 KN)^2 = (20 KN)^2 + (15 KN)^2
625 KN^2 = 400 KN^2 + 225 KN^2
625 KN^2 = 625 KN^2
Since both sides of the equation are equal, we have proven that the forces are acting at a right angle.
Hence, we can conclude that the forces F1 and F2 are acting at a right angle.