Question

A \(15 \mathrm{~N}\) force acts at \(35^{\circ}\) to the \(x\) axis. Resolve the force into forces in the \(x\) and \(y\) directions.

Short answer

Answer: The x-axis component of the force is approximately 12.29 N, and the y-axis component is approximately 8.60 N.

Step-by-step solution

Identify known values

We are given the following information: - Force magnitude = \(15 \mathrm{~N}\) - Angle with the x-axis = \(35^\circ\)

Resolve force into x-axis component

To resolve the force into its x-axis component, we can take the horizontal component of the force. This can be done using the cosine of the angle with the x-axis. The x-axis component of the force \(F_x\) can be calculated as: \(F_x = F \cdot \cos{\theta}\) where \(F\) is the force magnitude, and \(\theta\) is the angle with the x-axis. Plugging in the given values, we get: \(F_x = 15 \cdot \cos{35^\circ}\) \(F_x = 15 \cdot 0.8192\) \(F_x \approx 12.29 \mathrm{~N}\)

Resolve force into y-axis component

To resolve the force into the y-axis component, we can take the vertical component of the force. This can be done using the sine of the angle with the x-axis. The y-axis component of the force \(F_y\) can be calculated as: \(F_y = F \cdot \sin{\theta}\) where \(F\) is the force magnitude, and \(\theta\) is the angle with the x-axis. Plugging in the given values, we get: \(F_y = 15 \cdot \sin{35^\circ}\) \(F_y = 15 \cdot 0.5736\) \(F_y \approx 8.60 \mathrm{~N}\)

Summarize the results

The force \({15\ \mathrm{N}}\) acting at an angle of \({35^\circ}\) to the x-axis can be resolved into the following components: - x-axis component: \(F_x \approx 12.29\ \mathrm{N}\) - y-axis component: \(F_y \approx 8.60\ \mathrm{N}\)

Key concepts

Force Components

When a force acts at an angle, it can be broken down into two perpendicular components: one along the x-axis (horizontal) and the other along the y-axis (vertical). This process is known as resolving the force into its components.

• The horizontal component describes how much of the force acts in the horizontal direction.
• The vertical component describes how much of the force acts in the vertical direction.

Understanding force components is crucial in physics because it allows us to analyze the effects of forces more easily. This method comes in handy for various applications, such as understanding the motion of an object on an inclined plane or analyzing forces in structures like bridges.

Trigonometry in Physics

Trigonometry plays a vital role in physics, especially when dealing with forces. When a force is applied at an angle, trigonometric functions can help determine its components.

Using Cosine and Sine

Trigonometry involves using functions like cosine and sine to find the lengths of sides in a right triangle. In the context of resolving forces:

• To find the horizontal component ( \( F_x \) ), we use the cosine function: \( F_x = F \cdot \cos{\theta} \)
• To find the vertical component ( \( F_y \) ), we use the sine function: \( F_y = F \cdot \sin{\theta} \)

These relationships arise because in a right triangle formed with the force vector, the adjacent side (x-component) uses cosine, and the opposite side (y-component) uses sine of the angle.

Angle of Force Projection

The angle at which a force is applied greatly affects the magnitude of its components along the coordinate axes. This angle is known as the angle of force projection. Understanding this concept is crucial when analyzing forces in physics, especially in vector resolution.

• The larger the angle from the horizontal axis, the larger the vertical component relative to the horizontal one.
• A smaller angle leads to a larger horizontal component and a smaller vertical component.

It's essential to correctly measure and use this angle when calculating force components, as even a small deviation can lead to significant errors in determining the individual x and y components of the force. Recognizing the role of angle helps in accurately solving physics problems that involve forces and motion.