Problem 55 - Algebra and Trigonometry

Question

A wagon is pulled along level ground by exerting a force cof 40 pounds on a handle that makes an angle of \(32^{\circ}\) with the horizontal. How much work is done pulling the wagon 100 feet? Round to the nearest foot-pound.

Step-by-Step Solution

Answer

The work done pulling the wagon 100 feet can be found by following the above steps, which involve finding the horizontal component of the force and then using that force to find the work done.

Step 1: Convert from Degrees to Radians

Firstly, convert the angle from degrees to radians as we will be using trigonometric functions which take arguments in radians. The conversion rate is \( \frac{\pi}{180} \). So, \( \theta = 32^{\circ} = 32 * \frac{\pi}{180} \) radians.

Step 2: Calculate the Horizontal Force

The horizontal component of the force can be found using the equation \( F_{H} = F_{total} * \cos(\theta) \). Substituting in the known values gives us: \( F_{H} = 40 \text{ pounds} * \cos(32 * \frac{\pi}{180}) \).

Step 3: Compute the Work Done

Having found the effective force, we compute the work done by this force over a distance of 100 feet. Work, \( W \), is computed as the product of the force and the displacement, which gives us: \( W = F_{H} * 100 \text{ feet} \).

Key Concepts

Converting Degrees to Radians

Understanding how to switch between degrees and radians is crucial whenever trigonometric functions are involved, as these functions in the realm of physics typically require angles to be expressed in radians. This conversion is made simple by the relationship that reflects how the two units measure angles differently: a full circle is 360 degrees or \(2\pi\) radians. Therefore, to convert degrees to radians, we multiply by the constant factor \(\frac{\pi}{180}\).For example, a 32-degree angle converted into radians is calculated as \(32^\circ * \frac{\pi}{180}\) radians. This step is fundamental, as all subsequent calculations concerning trigonometric functions will rely on this conversion for accuracy.

Horizontal Force Calculation

When a force is applied at an angle, not all of that force is effective in moving an object in the desired direction. To identify the effective component, in this case, the horizontal component, we use the cosine trigonometric function. For a force \( F \) applied at an angle \( \theta \) to the horizontal, the horizontal component \( F_{H} \) is given by \( F_{H} = F * \cos(\theta) \).
Considering the cosine function provides the ratio of the adjacent side over the hypotenuse in a right-angled triangle, applying it to our scenario with the force and angle provided, gives us the force that's actually pulling the wagon forward. The accuracy of the horizontal force value is essential to calculate the work done accurately.

Trigonometric Functions in Physics

In physics, trigonometric functions help us resolve forces, analyze wave patterns, and solve problems involving periodic phenomena. The sine, cosine, and tangent functions are particularly important in dynamics where forces are applied at angles.

Trigonometry in Force Decomposition

Trigonometric functions enable us to break down a force vector into its horizontal and vertical components. This process, known as force decomposition, often involves the use of sine for the vertical component and cosine for the horizontal component. These functions are based on the relationship between the sides of a right triangle and the angles within it.
By understanding the properties of trigonometric functions, we can predict and calculate the effects of forces in a variety of physical contexts, from the simple motion of a wagon to the complex orbits of celestial bodies.

Work-Energy Principle

The work-energy principle is a fundamental concept that relates the work done on an object to its energy change. In physics, work \( W \) is defined as the product of the force \( F \) exerted on an object and the displacement \( s \) of the object in the direction of the force. It's expressed by the equation \( W = F * s \).

Calculating Work with Horizontal Force

When we consider an object being moved horizontally, like a wagon, the work done is specifically concerned with the horizontal component of the force. After calculating this component, we can apply the formula to determine the total work done in moving the object over a specified distance.
The concept links directly to energy, as work done on an object results in a transfer of energy, either adding to the kinetic energy of the object or overcoming forces such as friction. Understanding this principle helps students grasp how forces cause changes in the motion and energy of objects.