Introduction: In the realm of physics, understanding the equilibrium of forces and the resultant vector is fundamental to analyzing the interactions between different forces. This article delves into the problem of determining the relationship between forces P and Q when their resultant vectors are at right angles to each other, using the principles of trigonometry and vector addition.

Problem Definition

The problem at hand involves two forces, P and Q, acting on a particle. Their lines of action are inclined at 30° to each other. Initially, if Q acts in the opposite direction, the resultant force is R1. The challenge is to prove that if the lines of action of R and R1 are at right angles, then Q and P are equal, i.e., Q P. The article will also explore the related ratio between the magnitudes of P and Q.

Visualising Forces on a Coordinate System

To better visualize the forces, let us plot them on a X-Y chart with the particle at the origin. The force Q is represented as a vector OQ along the X-axis, and the force P is represented as a vector OP, which makes an angle of 30° with the X-axis.

When Q acts in the opposite direction, it can be represented as the vector -Q, which points along the X-axis in the negative direction. The angle between P and -Q is 150° (180° - 30°).

Proving the Relationship Between R and R1

Let's start by considering vectors R and R1. The resultant vector R when P and Q act together can be denoted as:

R2 P2 Q2 - 2PQ cos(30°) (1)

When Q acts in the opposite direction, the resultant vector R1 can be denoted as:

R12 P2 Q2 2PQ cos(30°) (2)

By solving these equations, we find that if P Q, the lines of action of R and R1 are at a right angle. This is because:

Therefore, the ratio of Q to R is:

Q/R 0.7195.

Further Simplification Using Dot Product

Knowing that vectors PQ and P - Q are perpendicular, their dot product must be zero:

(PQ) ? (P - Q) 0

Which simplifies to:

P2 - Q2 0

Thus, P Q, leading to the conclusion that R and R1 are the same vector when P Q.

Exploring the Triangular Relationships

Considering the angles formed by the vectors in the first quadrant, it can be seen that the angles are 150°, 15°, and 15°. Therefore, the relationship between the magnitudes of Q and R can be derived as:

R/2Q cos(15°)

Which gives us:

Q/R 0.5176.

Alternatively, we can derive the angle subtended by R and R1 using the tangent function and vector orientation. When the resultant vectors are at right angles, the angle A is:

tanA Q/(2P√3Q)

And the angle B when Q is inverted:

tanB Q/(2P - √3Q)

For right angles, tanAtanB 1, leading to the equation:

Q2/4P2 - 3Q2 4P2 - 2Q2/4P2 - 3Q2

Simplifying, we get:

4P2 2Q2

Therefore, Q P√2.

The final ratio of R to Q is:

R/Q √3√2/[P√2] ≈ 1.767.

Conclusion

This analysis demonstrates the intricate relationship between vector forces when they are in equilibrium. By understanding the principles of vector addition, trigonometry, and the properties of angles, we can derive significant insights into the interactions between different forces in a system. The balance and harmony of forces, when aligned at specific angles, reveal elegant mathematical relationships that are fundamental to the study of physics.