Solution:

Given: Magnitude of two forces = Q and √2Q

Let the angle between two forces be θ.

We know that the formula for resultant of two forces is given by:

R = √(F1^2 + F2^2 + 2F1F2cosθ), where F1 and F2 are magnitudes of two forces and θ is the angle between them.

Given that the magnitude of the resultant of two forces is Q, we can write:

Q = √(Q^2 + (√2Q)^2 + 2Q(√2Q)cosθ)

Solving this equation, we get:

cosθ = -1/√2

Therefore, the angle between two forces is 135 degrees.

Explanation:

To understand how we arrived at the solution, let us look at the steps involved in solving the problem.

Step 1: Write the formula for resultant of two forces

The formula for resultant of two forces is given by:

R = √(F1^2 + F2^2 + 2F1F2cosθ)

where F1 and F2 are magnitudes of two forces and θ is the angle between them.

Step 2: Substitute the given values

In this step, we substitute the given values into the formula for resultant of two forces. We are given that the magnitudes of two forces are Q and √2Q. Therefore, we can write:

R = √(Q^2 + (√2Q)^2 + 2Q(√2Q)cosθ)

Simplifying this equation, we get:

R = √(Q^2 + 2Q^2 + 2Q^2cosθ)

R = √(5Q^2 + 2Q^2cosθ)

Step 3: Use the given information to solve for θ

We are given that the magnitude of the resultant of two forces is Q. Therefore, we can write:

Q = √(5Q^2 + 2Q^2cosθ)

Simplifying this equation, we get:

Q^2 = 5Q^2 + 2Q^2cosθ

cosθ = (Q^2 - 4Q^2)/2Q^2

cosθ = -1/√2

Step 4: Find the angle between two forces

We know that cosθ = adjacent/hypotenuse. Since cosθ = -1/√2, we can say that adjacent = -1 and hypotenuse = √2. Using the Pythagorean theorem, we can find the opposite side of the triangle, which is √3. Therefore, we can write:

tanθ = opposite/adjacent

tanθ = √3/-1

tanθ = -√3

Taking inverse tangent on both sides, we get:

θ = 135 degrees

Therefore, the angle between two forces is 135 degrees.

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