Problem 218 - Calculus Volume 1

Question

For the following exercises, find the work done. Find the work done when a constant force \(F=12\) lb moves a chair from \(x=0.9\) to \(x=1.1 \mathrm{ft}\) .

Step-by-Step Solution

Answer

The work done is 2.4 foot-pounds.

Step 1: Understanding the Formula for Work

The formula for work done by a force is \( W = F \, \cdot \, d \), where \( W \) is the work, \( F \) is the constant force, and \( d \) is the distance over which the force is applied.

Step 2: Calculating the Distance

The chair is moved from \(x = 0.9\) ft to \(x = 1.1\) ft. To find the distance \(d\), subtract the initial position from the final position: \(d = 1.1 - 0.9 = 0.2\) ft.

Step 3: Substitute Values into the Formula

Now that we have the force \(F = 12\) lb and the distance \(d = 0.2\) ft, substitute these into the work formula: \[ W = 12 \times 0.2 \]

Step 4: Calculate the Work Done

Perform the multiplication to find the work done: \[ W = 12 \times 0.2 = 2.4 \] Thus, the work done is \(2.4\) foot-pounds.

Key Concepts

Work Formula

In physics, the concept of work relates to how much energy is transferred when a force is applied over a distance. The fundamental formula for calculating work is given by the equation:

• \( W = F \cdot d \)

Where:

• \( W \) represents the work done,
• \( F \) is the constant force applied,
• \( d \) is the distance over which the force acts.

This formula illustrates that the work done is directly proportionate to the force and the distance. Work is maximized when a large force moves an object over a long distance.

Constant Force

A constant force refers to a force that remains unchanged in magnitude and direction during the interaction. In practice, it is crucial because it simplifies calculations, allowing us to use a straightforward multiplication when calculating work.

• In our example, a force of 12 pounds is applied.

This constancy means that, for the entire duration the force moves the chair, its value does not fluctuate. Thus, the force is easily implemented in the work formula without complications.

Distance Calculation

Calculating the distance over which a force acts is a critical step in determining work. We first identify the initial and final positions of the object.

• Initial Position: \( x = 0.9 \) ft
• Final Position: \( x = 1.1 \) ft

The formula to calculate the distance \( d \) is:

• \( d = x_{final} - x_{initial} \)

Substituting the values:
\( d = 1.1 - 0.9 = 0.2 \) ft
This simple subtraction provides the distance over which the constant force does work.

Foot-Pound Unit

In the realm of physics, the foot-pound is a unit of work or energy. It reflects the amount of work done when a one-pound force is applied to move an object one foot.

• This is a part of the Imperial system, commonly used in the United States.

When you calculate work like in the exercise, where the force moves an object a certain distance, the result is expressed in foot-pounds. In our example, the resulting 2.4 foot-pounds signifies the energy transferred into moving the chair the specified distance.