Question

How much work is performed in moving a box horizontally 10ft with a force of 20lb applied at an angle of \(10^{\circ}\) to the horizontal?

Short answer

The work done is approximately 197 ft-lb.

Step-by-step solution

Understanding the Problem

To find the work done when moving a box, we need to consider the component of the force that is applied in the direction of movement. The angle of the applied force affects how much of the force actually contributes to moving the box horizontally.

Identifying the Horizontal Component of the Force

The horizontal component of the force can be found using the cosine of the angle between the force and the horizontal. The formula we use is: \(F_{horizontal} = F \cdot \cos(\theta)\), where \(F\) is the force applied, and \(\theta\) is the angle of the force. Here, \(F = 20 \text{ lb}\) and \(\theta = 10^{\circ}\).

Calculating the Horizontal Force

Substitute the given values into the formula for the horizontal component: \[F_{horizontal} = 20 \cdot \cos(10^{\circ})\]. Use a calculator to find \(\cos(10^{\circ}) \approx 0.9848\) and then calculate \(F_{horizontal} = 20 \times 0.9848 \approx 19.696 \text{ lb}\).

Applying the Work Formula

Work is calculated as the product of the horizontal force and the distance moved: \(W = F_{horizontal} \cdot d\), where \(d = 10 \text{ ft}\).

Calculating the Work Done

Substitute the values of the horizontal force and the distance: \[W = 19.696 \cdot 10 = 196.96 \text{ ft-lb}\].

Rounding Off the Final Answer

The calculated work is approximately \(196.96\text{ ft-lb}\). For simplicity, this is often rounded to \(197 \text{ ft-lb}\).

Key concepts

Horizontal Force

When you apply a force to move an object, like a box, horizontally, we are specifically interested in the portion of that force that acts parallel to the direction of movement. This is known as the horizontal force. Even if you apply a force at an angle, only the component aligned with the horizontal direction will do the work on the box.

For instance, in our exercise, a force of 20 pounds is applied at a 10° angle to the horizontal plane. To calculate the work done, we need to determine the horizontal component of this force. This means we'll separate out just the part of the force that is pushing directly in the horizontal direction.

Cosine of an Angle

In trigonometry, the cosine of an angle is critical for finding the horizontal component of a force. The cosine function relates the angle of the force to the adjacent side of a right triangle, which, in this case, is the horizontal component.

By using the cosine, we can mathematically express this relationship as:
* The formula is: F_{horizontal} = F \times \cos(\theta).
* Here, \(F\) is the magnitude of the force, and \(\theta\) is the angle of the force above the horizontal axis.

Using a calculator, \(\cos(10^{\circ})\) approximates to 0.9848. This value, when multiplied by the force magnitude, gives us the correct horizontal force component that contributes to moving the object.

Calculating Work

Work in physics is about applying a force over a distance. When calculating work, it's important to use only the component of force in the direction of movement. For horizontal movement, this means using the horizontal force component.

The formula for work is:
* \(W = F_{horizontal} \times d\)
* Where \(W\) is the work done, \(F_{horizontal}\) is the horizontal component of the force, and \(d\) is the distance moved.

In our problem, substituting the calculated horizontal force (19.696 lb) and the distance (10 ft) into this formula gives us \(W = 19.696 \times 10 = 196.96\ ft-lb\) of work done in moving the box.

Force Components

Understanding force components is essential when a force is not perfectly aligned with the direction of motion. Every force applied at an angle can be broken down into components:

• The horizontal component (which we calculated using cosine) is the part that moves the object forward.
• The vertical component, which does not contribute to horizontal work but might influence other factors like support or resistance.

By focusing on the horizontal component, we ensure that all calculations for work only consider forces acting in the intended direction of movement. This makes the process of calculating work more straightforward and precise, particularly when forces are applied at atypical angles.