Problem 39 - Algebra and Trigonometry

Question

Components of a Force \(\quad\) A man pushes a lawn mower with a force of 30 lb exerted at an angle of \(30^{\circ}\) to the ground. Find the horizontal and vertical components of the force.

Step-by-Step Solution

Answer

The horizontal component is approximately 25.98 lb and the vertical component is 15 lb.

Step 1: Identifying the Components

The problem gives us a force of 30 lb exerted at an angle of \(30^{\circ}\) to the horizontal. We can split this force into two perpendicular components: a horizontal component and a vertical component.

Step 2: Using Trigonometry to Find the Horizontal Component

To find the horizontal component, we use the cosine function. The horizontal component \(F_x\) is given by:\[ F_x = F \cdot \cos(\theta) \]where \(F = 30\, \text{lb}\) and \(\theta = 30^{\circ}\). Therefore,\[ F_x = 30 \cdot \cos(30^{\circ}) \approx 30 \cdot 0.866 \approx 25.98\, \text{lb} \]

Step 3: Using Trigonometry to Find the Vertical Component

To find the vertical component, we use the sine function. The vertical component \(F_y\) is given by:\[ F_y = F \cdot \sin(\theta) \]where \(F = 30\, \text{lb}\) and \(\theta = 30^{\circ}\). Therefore,\[ F_y = 30 \cdot \sin(30^{\circ}) \approx 30 \cdot 0.5 \approx 15\, \text{lb} \]

Step 4: Summarizing the Results

The horizontal component of the force is approximately \(25.98\, \text{lb}\), and the vertical component is \(15\, \text{lb}\). These components describe how the force acts along the horizontal and vertical directions.

Key Concepts

Vector Components

When dealing with forces, especially those acting at an angle, it's essential to break them down into simpler parts, known as vector components. Imagine a force as an arrow pointing in a certain direction; this arrow can be divided into horizontal and vertical components. This division helps in understanding how the force acts in different directions.
This way, solving problems becomes more manageable.

• Horizontal Component (F_x): Represents the force that acts parallel to the ground.
• Vertical Component (F_y): Represents the force that acts perpendicular to the ground.

Decomposing forces into these components allows us to analyze the effects of force with greater clarity. This is a fundamental part of physics and engineering, where forces often act in multiple directions simultaneously.

Sine and Cosine Functions

Trigonometry, a branch of mathematics dealing with triangles, introduces us to the sine and cosine functions. These functions are crucial when breaking down a force into its components.
Let's see how they help us:

• Cosine Function: It is used to find the horizontal component of the force when the angle is given with respect to the horizontal direction. The cosine function relates the adjacent side (horizontal component) to the hypotenuse (total force).
• Sine Function: It is used to determine the vertical component, relating the opposite side (vertical component) of the angle to the hypotenuse.

In our exercise, given a force of 30 lb at an angle of 30°:

• The horizontal component is calculated using the cosine function: \(F_x = F \cdot \cos(\theta)\).
• The vertical component is calculated using the sine function: \(F_y = F \cdot \sin(\theta)\).

Knowing these functions enables a precise calculation of each component, helping us fully understand how forces are resolved in physics.

Force Analysis

Force analysis involves examining how forces influence objects, which is vital in understanding motion and mechanics. Analyzing forces helps us pinpoint exactly how they interact with objects and the consequent effects.
In our specific exercise, analyzing the lawn mower’s force has practical implications:

• We determine the effectiveness of the force in moving the mower forward, which is linked to the horizontal component \( (25.98\, \text{lb}) \).
• The vertical component \( (15\, \text{lb}) \) gives insight into any lifting or downward pressure exerted on the mower, indicating how much force is working against gravity.

This kind of analysis is not just theoretical but also applicable in real-world scenarios like sports, engineering, and design. It connects mathematical concepts to tangible outcomes, enhancing our problem-solving skills.