Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.
forces of 250 and 450 newtons, forming an angle of 85°

Verified step by step guidance

1. Identify the two forces and the angle between them. Here, the forces are 250 N and 450 N, and the angle between them is 85°.
2. Recall the formula for the magnitude of the resultant force when two forces act at an angle \( \theta \): \[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \]
3. Substitute the given values into the formula: \[ R = \sqrt{250^2 + 450^2 + 2 \times 250 \times 450 \times \cos(85^\circ)} \]
4. Calculate the cosine of the angle \( \cos(85^\circ) \) using a calculator or trigonometric tables.
5. Evaluate the expression inside the square root step-by-step, then take the square root to find the magnitude of the resultant force.

Key Concepts

Vector Addition of Forces

Forces acting at a point can be represented as vectors. The resultant force is found by vector addition, combining both magnitude and direction. Understanding how to add vectors graphically or analytically is essential to determine the overall effect of multiple forces.

Law of Cosines

The Law of Cosines relates the lengths of sides in a triangle to the cosine of one of its angles. It is used to find the magnitude of the resultant vector when two vectors form a known angle, by treating the forces as sides of a triangle and the resultant as the third side.

Angle Between Vectors

The angle between two forces affects the magnitude of their resultant. Knowing how to use the given angle in calculations, especially in the Law of Cosines formula, is crucial to accurately compute the resultant force magnitude.