Problem 81 - 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 ft/s.

Step 1: Calculate the Circumference of the Winch

To find the speed at which the load is rising, we need to first calculate the circumference of the winch. Since the radius of the winch is given as 2 \( \text{ft} \), we use the formula for the circumference of a circle: \[ \text{Circumference} = 2 \pi \times \text{radius} = 2 \pi \times 2 = 4 \pi \text{ ft} \]

Step 2: Determine the Distance the Load Rises in 8 Revolutions

Each revolution of the winch lifts the load by a distance equal to the circumference of the winch. Since the winch makes 8 revolutions, the total distance the load rises is:\[\text{Distance} = 8 \times 4 \pi \text{ ft} = 32 \pi \text{ ft} \]

Step 3: Calculate the Speed of the Load

The speed of the load is the total distance it rises divided by the time taken. The winch takes 15 seconds for 8 revolutions. Thus the speed \( v \) is:\[ v = \frac{32 \pi \text{ ft}}{15 \text{ s}} \approx \frac{100.53 \text{ ft}}{15 \text{ s}} \approx 6.7 \text{ ft/s} \]

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around the circle's edge. It's similar to finding the perimeter of a square or rectangle but for a round shape. To calculate the circumference, we use the formula: \( C = 2 \pi r \), where \( r \) represents the radius of the circle, and \( \pi \approx 3.14159 \) is a constant. In the case of a winch with a radius of 2 feet, the circumference is:\[ 2 \pi \times 2 = 4 \pi \text{ ft} \]. This tells us each complete turn of the winch unwinds 4\( \pi \) feet of rope. Understanding the circumference helps us calculate how many feet of rope are played out with each rotation of the winch.
This fundamental circle property is crucial when dealing with any round objects in trigonometry or geometry.

Revolution and Rotation

Revolution and rotation are key concepts in understanding circular motion. A revolution refers to a full 360-degree turn or spin around an axis, often used in contexts of mechanical parts, like wheels or winches. Every time the winch completes one revolution, it moves the load by the distance of the winch's circumference, which is 4\( \pi \) ft in this problem.
Understanding rotations also involves knowing how many complete spins occur over a given time.

• In our winch scenario, it makes 8 revolutions in 15 seconds.
• This means the rope is extended 8 times the circumference per minute.

Grasping revolutions is crucial for determining the total distance over multiple spins. This furthers our understanding of the winch's mechanics and helps us connect to how it lifts the load in real time.

Speed Calculation

Speed calculation is concerned with determining how fast an object moves. In contexts like this problem, speed is the rate at which the load rises due to the winch. The basic formula for speed is \( v = \frac{d}{t} \), where \( v \) is velocity or speed, \( d \) is distance, and \( t \) is time.
Let's explore how this applies here:

• The total distance the load moves is the product of the number of revolutions and the circumference: 8 revolutions \( \times 4 \pi \text{ ft} = 32 \pi \text{ ft} \).
• The time taken for these revolutions is 15 seconds.
• Substituting these into the speed formula yields: \[ v = \frac{32 \pi \text{ ft}}{15 \text{ s}} \approx 6.7 \text{ ft/s} \]

This result tells us the average speed at which the winch's load is lifted. Understanding speed calculations allows us to evaluate how quickly operations occur, central to both practical applications and theoretical studies in physics.