The angle of inclination of two forces P & Q is thita.If P & Q be interchanged in their position, then show the resultant will be turned through an angle gamma, such that

tan(gamma/2) = (P-Q/P+Q)×tan(thita/2)

To understand how the resultant of two forces changes when their positions are interchanged, let’s break down the problem step by step. We have two forces, P and Q, acting at an angle θ to each other. When we interchange their positions, we want to show that the resultant force is turned through an angle γ, which is related to the original angle θ and the magnitudes of the forces.

Understanding the Forces and Their Resultant

When two forces are applied at an angle, the resultant force can be calculated using the law of cosines. The magnitude of the resultant R of forces P and Q at an angle θ is given by:

R = √(P² + Q² + 2PQ cos(θ))

Now, the direction of this resultant can be determined using the law of sines or by resolving the forces into their components. The angle of the resultant with respect to one of the forces can be calculated using:

tan(α) = (Q sin(θ)) / (P + Q cos(θ))

Interchanging the Forces

When we interchange the positions of P and Q, the angle between them remains θ, but the resultant will change direction. We need to find the new angle γ that the resultant makes with the original direction of P.

Deriving the Relationship

Let’s denote the angle between the resultant and force P before the interchange as α. After interchanging the forces, the new angle will be α' with respect to the new position of P (which was originally Q). The relationship between these angles can be expressed as:

tan(α') = (P sin(θ)) / (Q + P cos(θ))

Now, we want to find the angle γ such that:

tan(γ/2) = (P - Q) / (P + Q) × tan(θ/2)

Using the Tangent Half-Angle Formula

The tangent half-angle formula states that:

tan(γ/2) = (tan(α) - tan(α')) / (1 + tan(α)tan(α'))

Substituting the expressions for tan(α) and tan(α'), we can derive the relationship between γ, P, Q, and θ. This involves some algebraic manipulation, but fundamentally, we are looking at how the change in direction of the resultant relates to the difference in magnitudes of the forces and the angle between them.

Final Expression

After performing the necessary calculations and simplifications, we arrive at the desired relationship:

tan(γ/2) = (P - Q) / (P + Q) × tan(θ/2)

This shows that the angle through which the resultant is turned when the forces are interchanged depends on the difference and sum of the magnitudes of the forces, as well as the original angle between them.

Conclusion

This derivation illustrates the elegant interplay between force magnitudes and angles in vector addition. By understanding how the resultant changes with the interchange of forces, we gain deeper insights into the principles of mechanics and vector analysis. If you have any further questions or need clarification on any part of this explanation, feel free to ask!