Problem 45 - Advanced Engineering Mathematics

Question

A sled is pulled horizontally over ice by a rope attached to its front. A20-lb force acting at an angle of \(60^{\circ}\) with the horizontal moves the sled \(100 \mathrm{ft}\). Find the work done.

Step-by-Step Solution

Answer

The work done is 1000 ft-lb.

Step 1: Understand the Problem Scenario

We are given a force of 20 lb acting at an angle of \(60^\circ\) to the horizontal and we need to calculate the work done in moving the sled horizontally by 100 feet.

Step 2: Recall the Formula for Work

The work done by a force when moving an object is calculated using the formula: \( W = F \cdot d \cdot \cos(\theta) \). Here, \( F \) is the magnitude of the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and the displacement direction.

Step 3: Identify the Values for the Variables

From the problem, we have: \( F = 20 \) lb, \( d = 100 \) ft, and \( \theta = 60^\circ \).

Step 4: Calculate the Work

Substitute the known values into the formula: \[W = 20 \cdot 100 \cdot \cos(60^\circ)\] Calculate \( \cos(60^\circ) = \frac{1}{2} \), then \( W = 20 \cdot 100 \cdot 0.5 = 1000 \).

Step 5: Conclusion

The work done by the force is \( 1000 \) foot-pounds.

Key Concepts

Calculating Work

"Work" in physics involves calculating how much effort is applied to move an object over a distance. It's not simply about applying a force—it's about applying that force in a way that moves your object. To do this, we use the formula:

• Work (W) = Force (F) \(\times\) Distance (\(d\)) \(\times\) \(\cos(\theta)\) \

This formula explains that work depends on three main components:

• The magnitude of the force applied.
• The distance the object is moved.
• The angle (\(\theta\)) at which the force is applied relative to the direction of movement.

In the problem of the sled, we used a force of 20 lb, applied at a 60-degree angle, to move the sled 100 feet across ice. Substitute these values into our formula to find how much work is performed.When the angle between the force and the direction of movement is taken into account using the cosine, it determines how much of the force contributes effectively to moving the object. This means not all of the 20-lb force is used, but only the component of it that is directed along the path of the sled.In this case, the cosine of 60° is 0.5, which effectively reduces our force for calculation purposes. Thus, the total work done becomes 1000 foot-pounds, thereby quantifying the effort expended in moving the sled with efficiency.

Force and Angle Relationship

Force is a vector—it has both a magnitude and a direction. When lifting or pushing an object, the angle at which you apply the force significantly affects how effective your effort will be. This is where trigonometry, especially the cosine function, becomes crucial in physics calculations.In the scenario with the sled, the force was not aligned with the direction of movement; it was at a 60-degree angle. Such angular misalignment requires us to break down the force into components, specifically, its horizontal component.The horizontal component of a force is calculated as:

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

This calculation tells us how much of the 20 lb actually contributes to moving the sled straight forward. Here, \(\cos(60°) = 0.5\), meaning only half of the total force acts in the horizontal direction.Thus, the force used effectively in the sled problem was 10 lb (half of 20 lb). Using only this component in our work calculation ensures accuracy, as only the force which directly moves the sled is counted. Understanding force components helps in designing and executing tasks that rely on precise force and angle application for the best efficiency and performance.

Physics Problem Solving

Problem-solving in physics involves breaking down complex scenarios into manageable components, allowing us to analyze and compute various physical phenomena effectively. Mastering physics problem-solving often means becoming comfortable with deconstructing each part of a problem to clearly understand the principles at play. Here's a basic approach:

• Read and Understand the Problem: Carefully identify what you are being asked to find, and make a note of all given values. Contextual understanding helps in visualizing the physical setup.
• Identify the Relevant Equations: Choose the correct formula(s) related to the problem. This involves connecting the relevant principles, like Newton’s laws or work-energy relationships, to your problem.
• Substitute Known Values: Input the values given in the problem into your formula. Ensure units are consistent when performing these calculations.
• Calculate and Conclude: Finally, perform the mathematical operations, interpret the results, and check if they make sense in the context.

In the sled problem example, this means recognizing the work formula, substituting the relevant force, distance, and angle values, and carefully calculating to find the work done. Incorporating these steps ensures that you approach physics problems methodically, leading to greater accuracy and a deeper understanding of the intricate force and motion interplay.