Problem 13 - Applied Physics

Question

A gardener pushes a mower a distance of \(90 \overline{0} \mathrm{~m}\) in mowing a yard. The handle of the mower makes an angle of \(40.0^{\circ}\) with the ground. The gardener exerts a force of \(35.0 \mathrm{~N}\) along the handle of the mower (Fig. 8.8). How much work does the gardener do in mowing the lawn?

Step-by-Step Solution

Answer

The gardener does approximately 24,120 J of work in mowing the lawn.

Step 1: Understand the Definition of Work

Work is defined as the product of the force applied and the displacement in the direction of the force. Mathematically, it is given by the formula:\[ W = F \times d \times \cos(\theta) \]where \( W \) is the work done, \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and the displacement.

Step 2: Analyze Given Values

From the problem, we know:- The force \( F \) exerted by the gardener is 35.0 N.- The distance \( d \) the mower was pushed is 900 m.- The angle \( \theta \) between the mower handle and the ground is 40.0 degrees.

Step 3: Calculate the Horizontal Component of the Force

Since work is done in the direction of the displacement, we need the component of the force in the horizontal direction. Use cosine to find this component:\[ F_{\text{horizontal}} = F \times \cos(\theta) \]Substitute the known values:\[ F_{\text{horizontal}} = 35.0 \times \cos(40.0^{\circ}) \]

Step 4: Calculate Work Done Using Horizontal Force

Now that we have the horizontal component of the force, use it to calculate the work done:\[ W = F_{\text{horizontal}} \times d \]Substituting the value of \( F_{\text{horizontal}} \), we calculate the work done.

Step 5: Perform the Calculations

First, find \( F_{\text{horizontal}} \):\[ F_{\text{horizontal}} = 35.0 \times \cos(40.0^{\circ}) \approx 26.8 \, \text{N} \]Now, calculate the work:\[ W = 26.8 \, \text{N} \times 900 \, \text{m} = 24120 \, \text{J} \]

Key Concepts

Force and Displacement

Force and displacement are key components in understanding work in physics. **Force** is the push or pull exerted on an object to change its motion. Think of it as how hard you are pushing or pulling something. In this problem, the gardener pushes with a force of 35.0 N along the mower's handle. It's important to identify this force because it directly impacts the work done.
**Displacement** refers to how far an object moves. It's not just about distance but the change in position. The gardener moves the mower 900 meters while mowing the lawn. This displacement is significant because work relies on how far the force moves the object. Without displacement, despite the force exerted, no work is done. For work to be done, both force and displacement must be present. Just imagine pushing on a wall; without displacement, despite the exerted force, no work is done!

Angle of Force Application

The angle of force application is crucial in determining how much work is actually done on the object. In our example, the angle between the mower handle and the ground is 40.0 degrees.
This angle changes the effective part of the force that contributes to moving the mower. Not all of the force applied will move the mower horizontally due to this angle.
### Finding the Horizontal Component- We use trigonometry, specifically the cosine function, to determine how much of the total force contributes to displacement.- The formula is: \[ F_{\text{horizontal}} = F \times \cos(\theta) \] - Using this relationship, only part of the 35.0 N force pushes the mower along the ground. In these scenarios, understanding the angle helps us figure out how much force is really moving the object in the intended direction. This is necessary to calculate work accurately.

Work Calculation Formula

Work in physics is described through a specific formula that connects force, displacement, and the angle of force application: \( W = F \times d \times \cos(\theta) \).
Let's break down this formula: - **\( W \)** represents the work done.- **\( F \)** is the force applied.- **\( d \)** is the displacement of the object.- **\( \theta \)** is the angle between the force and the direction of displacement.By using this formula, we can calculate the work done by substituting the known values:- Inserting the force (35.0 N), displacement (900 m), and the cosine of the angle (40.0 degrees), gives: \[ W = 35.0 \, \text{N} \times 900 \, \text{m} \times \cos(40.0^{\circ}) \] - The cosine of 40 degrees is approximately 0.766, so the horizontal force becomes 26.8 N. Finally, the amount of work done is: \( W = 26.8 \, \text{N} \times 900 \, \text{m} = 24120 \, \text{J} \).
This formula is an essential tool for solving physics problems involving work, ensuring you consider all necessary components accurately.