Problem 54 - Contemporary Precalculus

Question

A child pulls a wagon along a level sidewalk by exerting a force of 18 pounds on the wagon handle, which makes an angle of \(25^{\circ}\) with the horizontal. How much work is done in pulling the wagon 200 feet?

Step-by-Step Solution

Answer

Answer: The work done is approximately 3260 foot-pounds.

Step 1: Identify the given values

From the exercise, we know that: Force, F = 18 pounds Angle, θ = \(25^{\circ}\) Distance, d = 200 feet;

Step 2: Calculate the horizontal component of the force

To find the horizontal component of the force, we need to use the angle given and the formula for the horizontal component of the force: Horizontal Force = F × cos(θ) We will use the above formula to find the horizontal force exerted by the child to pull the wagon. Horizontal Force = 18 pounds × cos(25°) Using a calculator, we get Horizontal Force ≈ 16.3 pounds;

Step 3: Calculate the work done in pulling the wagon

To calculate the work done in pulling the wagon, we can use the following formula: Work = Horizontal Force × Distance Now, using the values we calculated in the previous steps: Work = 16.3 pounds × 200 feet Work ≈ 3260 foot-pounds So the work done in pulling the wagon 200 feet is approximately 3260 foot-pounds.

Key Concepts

Horizontal Force

When you apply a force at an angle to the direction of movement, not all of that force contributes to the movement itself. The horizontal force is the part of the total force that actually moves the object in a horizontal direction. When calculating work, it's important to use the horizontal component of the force.

To find the horizontal force, you multiply the total force by the cosine of the angle of inclination. In this case, the child applies a force of 18 pounds at an angle of 25 degrees.

• The formula to calculate the horizontal force is: \[ \text{Horizontal Force} = F \times \cos(\theta) \]

Using the given values, horizontal force becomes approximately 16.3 pounds. It implies this is the effective force used to pull the wagon along the level sidewalk.

Angle of Inclination

The angle of inclination plays a crucial role in determining how much of the applied force actually contributes to the horizontal movement. An angle is formed between the direction of the force applied and the horizontal plane.

• In our problem, this angle is 25 degrees.

The part of the force contributing to motion is influenced by this angle because it affects how much force is directed towards the motion path versus elsewhere.

For angles less than 90 degrees, a portion of the force is always directed horizontally, calculated using the cosine function. As the angle increases up to 90 degrees, more force is wasted vertically and less is directed horizontally, which is why understanding this concept is vital in calculating work.

Distance

Distance is a key factor in determining the total work done. In physics, the distance refers to how far an object moves in the direction of the force applied. In the given problem, the wagon is pulled a distance of 200 feet.

Distance is directly proportional to the work done, meaning the greater the distance an object moves, the more work is required. However, note that this concept applies only as long as the force is applied consistently over that entire distance.

• The object needs to travel in a straight line for this relationship to hold true directly in our calculations.

Understanding how distance interacts with force helps in clearly comprehending the calculation of work output.

Calculating Work

Work is a measure of the energy transfer when a force moves an object over a distance, making it an essential concept in physics. The formula to calculate work is:

• \[ \text{Work} = \text{Horizontal Force} \times \text{Distance} \]

This calculation shows how energy is used to move an object from one place to another.

In our scenario, the horizontal force calculated earlier (16.3 pounds) is multiplied by the distance (200 feet) the wagon is moved, yielding approximately 3260 foot-pounds of work. This result indicates the amount of energy the child needs to exert to pull the wagon across that distance under the given conditions.

By understanding each element in this formula, students can better grasp how forces translate into movements and mechanical work.