Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

A tornado's core A simple model of the core of a tornado is a right circular cylinder that rotates about its axis. If a tornado has a core diameter of 200 feet and maximum wind speed of \(180 \mathrm{mi} / \mathrm{hr}\) (or \(264 \mathrm{ft} / \mathrm{sec}\) ) at the perimeter of the core, approximate the number of revolutions the core makes each minute.

Step-by-Step Solution

Answer

The tornado's core makes approximately 25 revolutions per minute.

Step 1: Convert Diameter to Radius

The diameter of the tornado's core is 200 feet. To find the radius, divide the diameter by 2.\[ \text{Radius} = \frac{200}{2} = 100 \text{ feet} \]

Step 2: Understand Wind Speed at Perimeter

The wind speed at the perimeter of the core is given as 264 feet per second. This means that any point on the edge of the core moves 264 feet in one second.

Step 3: Calculate the Circumference

The circumference of the circle (perimeter of the core) can be calculated using the formula for the circumference of a circle:\[ \text{Circumference} = 2\pi r = 2\pi \times 100 = 200\pi \text{ feet} \]

Step 4: Determine Revolutions per Second

A full revolution is equal to moving a distance equivalent to the circumference of the circle. To find how many revolutions per second (RPS) the perimeter makes:\[ \text{RPS} = \frac{\text{Wind Speed}}{\text{Circumference}} = \frac{264}{200\pi} \approx 0.42 \text{ revolutions per second} \]

Step 5: Convert Revolutions per Second to Revolutions per Minute

To find revolutions per minute (RPM), multiply the revolutions per second by 60 (since there are 60 seconds in a minute):\[ \text{RPM} = 0.42 \times 60 = 25.2 \approx 25 \text{ revolutions per minute} \]

Step 6: Round the Result

Rounding 25.2 to the nearest whole number gives us approximately 25 revolutions per minute.

Key Concepts

Tornado Core

The core of a tornado is a fascinating phenomenon that can be modeled as a right circular cylinder. In simple terms, imagine a large spinning tube made by the tornado as it touches the ground.

The flow inside this core is highly rotational, which means that the air tends to spin around the axis of the tornado. This rotation is fastest at the perimeter or edge of the core, which causes intense winds that define the tornado's strength.

Understanding the core's structure is vital for calculating important parameters like wind speed and the tornado's angular motion. The diameter is key as it helps define the core's size and is essential in determining how the tornado interacts with the environment around it.

Angular Velocity

Angular velocity is a measure of how quickly something spins around an axis. In the case of a tornado, it tells us how fast the core rotates.

This velocity is different from linear velocity, as it specifically refers to the rate of rotation and not how far a point moves in a straight line. Angular velocity () is usually calculated using the formula: \[ \omega = \frac{ ext{Linear Speed}}{ ext{Radius}} \]

For our tornado model, knowing the wind speed at the core's perimeter allows us to compute how fast the core itself rotates. The faster the angular velocity, the more fierce the tornado can be.

Wind Speed

Wind speed at the edge of a tornado's core is crucial for understanding the tornado's power. It is given as 264 feet per second in our exercise, which is the speed at which air moves along the perimeter.

Wind speed determines the kind of damage a tornado can cause, with higher speeds resulting in more violent storms. It provides direct insight into the tornado's intensity and helps meteorologists classify its strength on various scales.

This specific speed measurement serves as the basis for calculating other important characteristics such as angular velocity and revolutions per minute. Understanding these dynamics reveals why the tornado's edge is so destructive.

Revolutions per Minute

Revolutions per minute (RPM) is a measure of the number of complete rotations the tornado core makes in one minute.

To find the RPM, it's first necessary to calculate revolutions per second by dividing the wind speed by the core's circumference.

Multiplying this figure by 60 converts it from seconds to minutes, providing RPM. In our exercise, this calculation shows approximately 25 revolutions per minute.

Understanding RPM is important as it gives a practical sense of how rapidly the tornado core spins. This spin rate correlates directly with the tornado's destructive potential, as faster revolving cores tend to have higher wind speeds and more significant impact.