Given:
- Magnitude of force 1 (p) = p
- Magnitude of force 2 (q) = q
- Angle between the forces (θ) = 60 degrees
- Resultant of the forces (R) = √7q

To find:
- P/Q

Explanation:

1. Resolving Forces:
When two forces act at an angle, they can be resolved into two perpendicular components. Let's resolve the forces p and q into their respective components.

- Force p can be resolved into two components:
- p₁, which is parallel to the resultant force R
- p₂, which is perpendicular to the resultant force R

- Force q can also be resolved into two components:
- q₁, which is parallel to the resultant force R
- q₂, which is perpendicular to the resultant force R

2. Relation between Components:
The components parallel to the resultant force R add up, while the components perpendicular to the resultant force R cancel each other out.

Since the resultant of the forces is √7q, we can write the following equations:

- p₁ + q₁ = √7q (Equation 1)
- p₂ + q₂ = 0 (Equation 2)

3. Finding the Ratio:
To find the ratio P/Q, we need to express p and q in terms of their components.

From Equation 1, we can solve for p₁:

p₁ = √7q - q₁

From Equation 2, we can solve for p₂:

p₂ = -q₂

Now, we can substitute the values of p₁ and p₂ into the equation P/Q:

P/Q = (p₁ + p₂) / q

Substituting the values:

P/Q = (√7q - q₁ - q₂) / q

4. Simplifying the Ratio:
Since q₁ and q₂ are the components of force q perpendicular to the resultant R, we can relate them using trigonometry.

From the right-angled triangle formed by q, q₁, and q₂, we have:

sin(60) = q₁ / q

sin(60) = √3 / 2

q₁ = (√3 / 2) * q

Similarly, q₂ = (1 / 2) * q

Substituting the values of q₁ and q₂ into the P/Q equation:

P/Q = (√7q - (√3 / 2) * q - (1 / 2) * q) / q

P/Q = (√7 - (√3 / 2) - (1 / 2))

5. Final Ratio:
Simplifying the expression:

P/Q = (√7 - (√3 / 2) - (1 / 2))

P/Q = (√7 - √3 - 1) / 2

Hence, the ratio P/Q is (√7 - √3 - 1) / 2.

Thanks