Problem 77 - Algebra and Trigonometry

Question

Winch A winch of radius 2 \(\mathrm{ft}\) is used to lift heavy loads. If the winch makes 8 revolutions every 15 \(\mathrm{s}\) , find the speed at which the load is rising.

Step-by-Step Solution

Answer

The load is rising at approximately 6.7 feet per second.

Step 1: Find the Circumference of the Winch

The circumference of a circle (or in this case, the winch) is given by the formula \( C = 2 \pi r \), where \( r \) is the radius. Given the winch's radius is 2 \( \text{ft} \), the circumference is \( C = 2 \pi \times 2 = 4\pi \ \text{ft} \).

Step 2: Calculate the Distance per Revolution

Each time the winch makes one full revolution, it pulls up a length of rope equal to its circumference. Therefore, the distance the load rises per revolution is \( 4\pi \ \text{ft} \).

Step 3: Determine the Total Distance for 8 Revolutions

If the winch makes 8 revolutions, the total distance the load rises is \( 8 \times 4\pi = 32\pi \ \text{ft} \).

Step 4: Calculate the Time for 8 Revolutions

Given that the winch makes 8 revolutions every 15 seconds, the time taken is 15 seconds.

Step 5: Find the Speed of the Load

The speed is the total distance divided by the total time. Thus, the speed at which the load rises is \( \frac{32\pi}{15} \ \text{ft/s} \).

Step 6: Simplify the Speed

Simplify the expression: \( \frac{32\pi}{15} \ \text{ft/s} \approx 6.7 \ \text{ft/s} \).

Key Concepts

Understanding Circle Circumference

The circle circumference is an essential concept in trigonometry, which helps us determine the perimeter of circles. For any circle or circular object, you can calculate its circumference using the formula \( C = 2\pi r \). Here, \( C \) is the circumference and \( r \) is the radius of the circle.
In the case of the winch with a radius of 2 feet, we apply this formula to find its circumference. It's crucial to remember that \( \pi \) is a constant, approximately equal to 3.14159. When plugged into the formula, the circumference of the winch becomes \( 2 \pi \times 2 = 4\pi \ \text{ft} \).

• This means the edge of the winch measures \( 4\pi \ \text{ft} \) around.
• Every full turn of the winch equals the length of its circumference.

This understanding is crucial for determining how far the winch pulls up the load with each revolution.

Revolution Distance Mechanics

Revolution distance is directly tied to the circumference we calculated earlier. When a circular object, such as our winch, completes one full revolution, it covers a distance equal to its circumference.
For the winch in our exercise, each revolution raises the load by \( 4\pi \ \text{ft} \) since that's the length around the winch. This realization allows us to compute the total distance traveled for any number of revolutions.

• With 8 revolutions, the winch moves the load a total distance of \( 8 \times 4\pi = 32\pi \ \text{ft} \).
• Each revolution contributes independently to the total distance.

This calculation is fundamental for understanding how circular motion affects linear displacement over multiple revolutions.

Basic Speed Calculation Principles

Speed calculation in this context involves measuring how fast the load is raised by the winch. Speed is defined as the distance traveled per unit of time. The formula is simple: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
In our example, the total distance the load rises is \( 32\pi \ \text{ft} \) over a time span of 15 seconds.

• To find the speed, you divide the total distance by the total time: \( \frac{32\pi}{15} \ \text{ft/s} \).
• After simplification, this gives approximately \( 6.7 \ \text{ft/s} \).

This calculation shows us the efficiency of the winch in lifting loads, illustrating a key practical application of trigonometry and basic physics in real-world scenarios.