The two forces $2\sqrt{2}\text{N}\text{and}x\text{N}$ are acting at a point their resultant is perpendicular to $\hat{x}\text{N}$ and having magnitude of $\sqrt{6}\text{N}$ . The angle between the two forces and magnitude of x are

The problem involves determining the angle between two forces and the magnitude of the unknown force x, where the resultant force is perpendicular to F2 and has a magnitude of 6 N.

Step 1: Forces and Resultant

Given forces:

• F1 = 2√2 N
• F2 = xn 2 N

The resultant force R is perpendicular to F2 and has a magnitude of 6 N.

Step 2: Using the Pythagorean Theorem

Since the resultant force is perpendicular to F2, we can use the Pythagorean theorem:

R² = F1² + F2²

Substituting the values:

6² = (2√2)² + (xn 2)²

This simplifies to:

36 = 8 + (xn 2)²

Rearranging gives:

(xn 2)² = 36 - 8 = 28

Taking the square root:

xn 2 = √28 = 2√7

Step 3: Angle Between the Forces

The angle θ between the forces can be found using the resultant formula:

R² = F1² + F2² + 2F1F2cos(θ)

Substituting the known values:

36 = (2√2)² + (xn 2)² + 2(2√2)(xn 2)cos(θ)

With xn 2 = 2√7, this becomes:

36 = 8 + 28 + 8√14 cos(θ)

Simplifying:

36 = 36 + 8√14 cos(θ)

8√14 cos(θ) = 0

This implies cos(θ) = 0, so θ = 90°.

Step 4: Verify Perpendicularity

Since the resultant is perpendicular to F2, the angle between the forces must allow for this condition. The calculation confirms the angle is 90°, satisfying the condition of perpendicularity.

Final Answers

• The angle between the forces is 90°.
• The magnitude of xn 2 is 2√7 N.