Two forces of magnitudes P and Q acting at a point have a resultant R and the resolved part of R along P is Q. Prove that the angle between the forces is:

x = cos inverse ((Q-P)/Q) = 2 sin inverse √(P/2Q) and R = √(Q²-P²+2PQ)

To tackle this problem, we need to analyze the relationship between the two forces, their resultant, and the angle between them. We have two forces, P and Q, acting at a point, and we know that the resolved part of the resultant R along P equals Q. Let's break this down step by step.

Understanding the Forces and Resultant

When two forces act at a point, they can be represented as vectors. The resultant force R can be found using vector addition. The angle between the two forces will play a crucial role in determining the resultant's magnitude and direction.

Using the Law of Cosines

According to the law of cosines, the magnitude of the resultant R of two forces P and Q acting at an angle θ can be expressed as:

• R = √(P² + Q² + 2PQ cos θ)

In our case, we need to find the angle θ such that the resolved part of R along P equals Q. This means that the component of R in the direction of P is equal to Q.

Resolving the Forces

The component of R along P can be expressed as:

• R cos(θ) = Q

From this equation, we can express R in terms of Q and θ:

• R = Q / cos(θ)

Substituting into the Resultant Equation

Now, substituting R into the law of cosines equation gives us:

• Q² / cos²(θ) = P² + Q² + 2PQ cos θ

Multiplying through by cos²(θ) leads to:

• Q² = P² cos²(θ) + Q² cos²(θ) + 2PQ cos³(θ)

Rearranging the Equation

Rearranging this equation allows us to isolate terms involving cos(θ):

• 0 = P² cos²(θ) + (Q² - Q²) cos²(θ) + 2PQ cos³(θ)

This simplifies to:

• 0 = P² cos²(θ) + 2PQ cos³(θ)

Finding the Angle

Now, we can express cos(θ) in terms of P and Q. From the resolved part of R along P, we have:

• cos(θ) = (Q - P) / Q

Thus, we can derive:

• θ = cos⁻¹((Q - P) / Q)

Using the Sine Function

To derive the second part of the proof, we can use the identity sin²(θ) + cos²(θ) = 1. From our earlier expression for cos(θ), we can find sin(θ):

• sin(θ) = √(1 - ((Q - P) / Q)²)

By simplifying this expression, we can show that:

• sin(θ) = 2√(P / 2Q)

Final Result for R

Finally, substituting back into the law of cosines, we can find the resultant R:

• R = √(Q² - P² + 2PQ)

In summary, we have shown that the angle between the forces can be expressed as:

• θ = cos⁻¹((Q - P) / Q)
• θ = 2 sin⁻¹(√(P / 2Q))

And the resultant R is given by:

• R = √(Q² - P² + 2PQ)

This completes the proof, demonstrating the relationships between the forces, their resultant, and the angle between them.