Understanding the X-Component of a Force Given Its Magnitude and Angle with the Y-Axis

Introduction to Forces and Trigonometry

When dealing with forces in mechanics, it's often necessary to break down the force into its components along different axes. In this article, we will explore how to find the x-component of a force when you know its magnitude and the angle it makes with the y-axis. This involves the application of trigonometry, specifically the cosine and sine functions.

Problem Statement

A force of 10N makes an angle of 30° with the y-axis. What will be the magnitude of its x-component?

Step-by-Step Solution

Using Trigonometric Functions

Given:

Force (F): 10N An angle with the y-axis (θ): 30°

First, we need to consider that the given angle is with respect to the y-axis to determine the appropriate trigonometric function for the x-component. Since we need to find the adjacent side (x-component) to the angle, we use the cosine function. However, we should note that the angle needs to be converted to the corresponding angle with respect to the x-axis.

Calculation Steps:

1. **Convert the Angle to X-Axis**

Since the angle with the y-axis is 30°, the angle with the x-axis would be 90° - 30° 60°.

2. **Use the Cosine Function**

Now, we use the cosine function to find the x-component.

Fx F · cos(60°)

Given F 10N, and cos(60°) 1/2:

Fx 10 N · cos(60°)

Fx 10 N · (1/2)

Fx 5 N

Verification

Another way to verify the same solution is to directly use:

Fx 10 N · sin(30°)

Given that sin(30°) 1/2:

Fx 10 N · (1/2)

Fx 5 N

Angle Considerations

Understanding that the angle is measured clockwise or counterclockwise from the y-axis is crucial. For instance, 30° clockwise from the y-axis is a 60° angle with the positive x-axis. If we consider a 30° angle from the negative y-axis, it would result in an angle of 150° from the positive x-axis, as cos(150°) -1/2.

For a 30° counterclockwise angle from the y-axis, the angle with the positive x-axis would be 120°, and the x-component would be negative:

Fx -10 N · cos(120°)

Fx -10 N · (-1/2)

Fx 5 N (towards negative x direction)

Further Understanding of Trigonometric Ratios

To deepen the understanding, it's beneficial to memorize fundamental triangles for quick reference:

30°-60°-90° Triangle: The sides are in the ratio 1:2:√3. 45°-45°-90° Triangle: The sides are in the ratio 1:1:√2.

These triangles help in quickly calculating the sine and cosine values for these angles, which is particularly useful in mechanics and physics problems.

Conclusion

By using the correct trigonometric function and understanding angle conversion, it is straightforward to determine the x-component of a force given its magnitude and its angle with the y-axis. This knowledge is fundamental in applied mathematics and physics, providing a solid base for solving more complex problems in the field.