Problem: If two forces act in opposite direction then their resultant is 10N and if they act mutually perpendicular then their resulant is 50N.find the magnitude of both the forces?

Solution:

Let F1 and F2 be the magnitude of the two forces.

Case 1: Forces acting in opposite direction

Given, the resultant force is 10N.

Therefore, we can write:

F1 - F2 = 10N ...(1)

Case 2: Forces acting perpendicular to each other

Given, the resultant force is 50N.

Therefore, we can write:

F1^2 + F2^2 = 50N^2 ...(2)

Solving the equations (1) and (2) simultaneously:

From equation (1), we get:

F1 = F2 + 10N

Substituting the value of F1 in equation (2), we get:

(F2 + 10N)^2 + F2^2 = 50N^2

Simplifying, we get:

2F2^2 + 20F2 - 1500 = 0

Solving this quadratic equation, we get:

F2 = 25N or F2 = -30N

Since force can't be negative, we reject the solution F2 = -30N.

Therefore, F2 = 25N.

Substituting the value of F2 in equation (1), we get:

F1 - 25N = 10N

Therefore, F1 = 35N.

Hence, the magnitude of the two forces are F1 = 35N and F2 = 25N.

Explanation:

When two forces act in opposite direction, their resultant is the difference between the two forces. In this case, we have F1 and F2 acting in opposite direction, and their resultant is 10N.

When two forces act perpendicular to each other, their resultant is given by the Pythagorean theorem. In this case, we have F1 and F2 acting perpendicular to each other, and their resultant is 50N.

To solve for the magnitude of the two forces, we use simultaneous equations. We first use equation (1) to solve for F1 in terms of F2. Then we substitute the value of F1 in equation (2) and simplify to get a quadratic equation in F2. Solving this equation gives us the magnitude of F2, and we can use equation (1) again to solve for the magnitude of F1.

Answer:

Therefore, the magnitude of the two forces are F1 = 35N and F2 = 25N.