Problem 18 - Applied Physics

Question

Two tractors tow a barge through a canal; each tractor uses a towrope that makes an angle of \(21^{\circ}\) with the bank of the canal. If the tension in each rope is \(12,000 \mathrm{~N}\), how much work is done in moving the barge \(550 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

The work done is approximately \(12,323,520\) Joules.

Step 1: Understand the Problem

The problem involves two tractors towing a barge each with an angle of \(21^{\circ}\) to the bank, a tension of \(12,000\, \text{N}\) in each rope, and a distance of \(550\, \text{m}\). We need to find the work done in moving the barge.

Step 2: Identify the Relevant Formula

Work done is calculated by the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \( W \) is work, \( F \) is the force, \( d \) is the distance, and \( \theta \) is the angle between the force and direction of motion.

Step 3: Calculate the Effective Force

Since each tractor's force adds up, calculate the horizontal component of each force using \( F_{\text{horizontal}} = F \cdot \cos(\theta) \), therefore \( F_{\text{horizontal}} = 12,000 \cdot \cos(21^{\circ}) \). Multiply by 2 since there are two tractors: \( F_{\text{total}} = 2 \times 12,000 \cdot \cos(21^{\circ}) \).

Step 4: Calculate Work Done

Substitute \( F_{\text{total}} \) and \( d = 550\, \text{m} \) into the work formula: \( W = F_{\text{total}} \cdot d \). Calculate to find \( W = 2 \times 12,000 \times \cos(21^{\circ}) \times 550 \).

Step 5: Final Calculation

Compute the values: \( \cos(21^{\circ}) \approx 0.9336 \). Thus, \( F_{\text{total}} \approx 2 \times 12,000 \times 0.9336 \). The total force equals \( 22,406.4 \text{ N} \) approximately. Therefore, \( W \approx 22,406.4 \times 550 \approx 12,323,520 \text{ J (joules)} \).

Key Concepts

Force Components

When dealing with physics problems involving forces applied at angles, it's essential to understand the concept of force components. Imagine a force as an arrow pointing in a particular direction. To better analyze how this force acts, we can break it down into components. This involves splitting the force into two perpendicular directions: horizontal and vertical.

In the case of the barge being towed by the tractors, each rope exerts a force at an angle of \(21^{\circ}\) to the canal bank. Therefore, not all of the tensile force (or tension in the rope) is used to pull the barge directly along the canal. Only the horizontal component of the force contributes to moving the barge along its path.

To find this horizontal component, we use the formula:

• \( F_{\text{horizontal}} = F \cdot \cos(\theta) \)

where \( F \) is the magnitude of the total force (tension in this case), and \( \theta \) is the angle of the force. In this problem, each tractor exerts a force of \(12,000 \, \mathrm{N}\), so the horizontal component for one tractor is \(12,000 \cdot \cos(21^{\circ})\). Since there are two tractors, we multiply this component by two to find the total effective force pulling the barge along the canal.

Trigonometry in Physics

Trigonometry plays an important role in physics, helping us understand how different components of forces act in multi-dimensional spaces. When a force is applied at an angle, as is common in many real-world situations, we often use trigonometry to analyze its effects.

The problem presents an angle of \(21^{\circ}\), which we use to determine the amount of force that actually moves the barge forward. The tool for this conversion is the trigonometric function cosine. Cosine helps us project the force vector onto the horizontal axis.

• The function is defined as \( \cos(\theta) \), where \( \theta \) is the angle between the force direction and the horizontal axis.

By multiplying the tension by \( \cos(21^{\circ}) \), we find the fraction of the force acting in the desired direction.

This use of trigonometry ensures that we only focus on the component of the force that effectively contributes to moving the barge, ignoring the vertical component, which doesn't affect horizontal movement.

Problem Solving in Physics

Successfully solving physics problems often requires a systematic approach. Let’s break it down step by step, using our barge and tractors scenario as an example.

First, understand the problem by clearly identifying what’s given—here, the tension, the angle, and the distance—and identifying what’s being asked, which is the work done.

Next, use relevant formulas. For work, the fundamental formula is:

• \( W = F \cdot d \cdot \cos(\theta) \)

where \( W \) is work, \( F \) is the effective force in the direction of movement, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle of force application.

Then, always solve for the effective force first. For the barge, calculate \( F_{\text{total}} \) using the cosine of the angle to account for the horizontal contribution from both tractors.

Finally, apply these values to find the work done. Plug everything into the work formula and carefully calculate. Always double-check each computation! In this exercise, following these enjoyable systematic steps led us to the answer: approximately \(12,323,520\) joules of work were done in moving the barge.