Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

Verified step by step guidance

1. Identify the given values: the magnitudes of the two forces are \(F_1 = 692\) newtons and \(F_2 = 423\) newtons, and the magnitude of the resultant force is \(R = 786\) newtons.
2. Recall the formula for the magnitude of the resultant of two forces acting at an angle \(\theta\) between them: \(R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta}\)
3. Substitute the known values into the formula: \(786 = \sqrt{692^2 + 423^2 + 2 \times 692 \times 423 \times \cos \theta}\)
4. Square both sides to eliminate the square root: \(786^2 = 692^2 + 423^2 + 2 \times 692 \times 423 \times \cos \theta\)
5. Rearrange the equation to solve for \(\cos \theta\): \(\cos \theta = \frac{786^2 - 692^2 - 423^2}{2 \times 692 \times 423}\)

Key Concepts

Resultant of Two Forces

The resultant force is the single force that has the same effect as two or more forces acting together. When two forces act at a point, their resultant can be found using vector addition, which depends on the magnitudes of the forces and the angle between them.

Law of Cosines in Vector Addition

The law of cosines relates the magnitudes of two vectors and the angle between them to the magnitude of their resultant. It states that R² = A² + B² + 2AB cos(θ), where R is the resultant, A and B are the forces, and θ is the angle between them.

Solving for the Angle Between Forces

To find the angle between two forces given their magnitudes and resultant, rearrange the law of cosines formula to solve for cos(θ). Then use the inverse cosine function to determine the angle, ensuring the correct interpretation of the angle in the context of the problem.