Problem 42 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use vectors and dot products to study real- world problems. A child pulls a wagon along level ground. He exerts a force of 20 pounds on the handle, which makes a \(30^{\circ}\) angle with the horizontal. Find the work done in pulling the wagon 100 feet, to the nearest foot-pound.

Step-by-Step Solution

Answer

The work done in pulling the wagon is approximately 1732 foot-pounds.

Step 1: Calculate the Horizontal Component of Force

Use the angle of 30 degrees between the handle and the horizontal to determine the horizontal component of the force. The component of the force in the direction of motion (F horizontal) can be calculated using the cosine function: \(F_{horizontal} = F \cdot cos(\theta)\) where \(F = 20\, lb\) and \(\theta = 30^{\circ}\). According to the formula, \(F_{horizontal} = 20 \cdot cos( 30^{\circ})\).

Step 2: Evaluate cosine function

Evaluate the cosine function using a trigonometry table or a calculator. \(cos(30^{\circ})\) equals \(\frac{\sqrt{3}}{2}\). The horizontal force \(F_{horizontal}\) becomes \(F_{horizontal} = 20 \cdot \frac{\sqrt{3}}{2}\).

Step 3: Simplify the expression

Simplify the expression obtained above for \(F_{horizontal}\). The multiplication gives \(F_{horizontal} = 10\sqrt{3}\, lb\).

Step 4: Calculation of Work Done

Now, calculate the work done. By definition, work done (W) is given by the product of the force and the displacement (d), i.e., \(W = F_{horizontal} \cdot d\). Substituting the known values, \(W = 10\sqrt{3} \cdot 100\).

Step 5: Final Simplification

Simplify this last expression to obtain the final answer: \(W = 1000\sqrt{3}\, foot-pounds\). The answer can be rounded to the nearest foot-pound if needed. The square root of 3 is approximately 1.732, so the work done is approximately 1732 foot-pounds.

Key Concepts

Dot Product

The dot product is a fundamental operation used in vector mathematics. When dealing with vectors, the dot product helps us find how much one vector goes in the direction of another vector. In this context, the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them. You can calculate it using the formula:

• \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| \cdot |\mathbf{B}| \cdot \cos(\theta) \)

This operation is significant in physics, especially when calculating work done, as it allows us to break down forces into useful components. In our exercise, the force exerted by the child is at an angle, and to find the work done, we only care about the component in the direction of motion. Using the dot product, we can easily find this component by multiplying the force's magnitude by the cosine of the angle involved. It essentially converts the exerted force into an effective force along the desired direction.

Trigonometry

Trigonometry is the branch of mathematics that handles the relationships between angles and sides in triangles. It is particularly useful in physics and engineering for dividing vectors into components that align with useful directions. In our solved problem, trigonometry is used to resolve the force vector into components.Using trigonometric functions, like cosine, allows us to find the part of a vector that aligns with any direction. In this problem, we use the cosine function to find the horizontal component of the force. The formula applied is:

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

Where \( F \) is the magnitude of the force and \( \theta \) is the angle relative to the horizontal. Trigonometry tables or calculators usually help find the cosine of angles like \( 30^{\circ} \). Here, it resulted in \( \frac{\sqrt{3}}{2} \), allowing us to calculate the precise horizontal force that contributes to the work done on the wagon.

Work and Energy

Work and energy are core concepts in physics related to how forces lead to motion. Work is essentially the measure of energy transfer that occurs when an object is moved by a force. In simple terms, work is done when a force moves something in its direction, and it is numerically equivalent to the force's component in the direction of movement times the distance moved.In mathematical terms, work \( W \) is given by:

• \( W = F \cdot d \)

Where \( F \) is the component of force in the direction of movement and \( d \) is the distance over which the force is applied. The unit of work is typically the foot-pound or joule, depending on the measurement system. In the exercise problem, the child performs work by pulling the wagon 100 feet with a resolved horizontal force. This force and distance are multiplied to give the total work done, which, after calculating, approximates to 1732 foot-pounds.