Problem 99 - Precalculus : Building Concepts and Connections

Question

In one of the rides at an amusement park, you sit in a circular "car" and cause it to rotate by turning a wheel in the center. The faster you turn the whecl, the faster the car rotates. How far from the center of the car are you sitting if your car makes one revolution cuery 3 seconds and your lincar speed is 5 feet per second? Express your answer in feet.

Step-by-Step Solution

Answer

The distance from the center of the car is approximately \( \frac{15}{2\pi} \) or about 2.39 feet.

Step 1: Understand the problem and identify the given values

It is given that the linear speed of the car (v) is 5 feet/second and the time it takes to complete one revolution (T) is 3 seconds. We are required to find the radius (r) of the circle, which in this case is the distance from the center of the car where we're sitting. We are also aware that one complete revolution is equivalent to an angle of \(2\pi\) radians.

Step 2: Calculate the angular speed

Angular speed (\(\omega\)) is calculated as \(\omega = \frac{2\pi}{T}\). From the given values, we can substitute \(T = 3\) seconds to get \(\omega = \frac{2\pi}{3}\) radians per second.

Step 3: Solve for the radius

We can use the fact that the linear speed is the product of the radius and the angular speed (\(v = r\omega\)). So we rearrange the equation to solve for the radius \(r = \frac{v}{\omega}\). Substituting values, we have \(r = \frac{5}{\frac{2\pi}{3}}\). This simplifies to \(r = \frac{15}{2\pi}\) feet.

Key Concepts

Linear Speed

In circular motion, linear speed refers to how fast you're moving along the path of the circle. It is typically measured in units such as feet per second or meters per second. Imagine you're on a carousel; the linear speed tells you how quickly you're traveling around the circle. This is different from simply spinning in place or rotating, where nothing physically moves along a path.

To calculate your linear speed in a circular path, use the formula:

• \(v = r \cdot \omega\)

Where \(v\) is your linear speed, \(r\) is the radius of the circle (distance from the center), and \(\omega\) is the angular speed in radians per second.

In our example scenario, with a linear speed of 5 feet per second, understanding this helps set the groundwork for calculating other important factors like angular speed and the radius. This concept is crucial when solving problems involving circular paths.

Angular Speed

Angular speed is like the "rotating speed" of an object moving in a circle. It describes how fast the object is spinning or how quickly it's covering its circular path. Angular speed is often measured in radians per second. A full revolution of a circle is equivalent to \(2\pi\) radians.

To find angular speed, use the formula:

• \(\omega = \frac{2\pi}{T}\)

Where \(\omega\) is the angular speed and \(T\) is the time taken to complete one full revolution.

For instance, in the example problem, where one complete revolution takes 3 seconds, you've found the angular speed to be \(\frac{2\pi}{3}\) radians per second. This calculation is essential for understanding the motion's rotational aspect and linking your known linear speed to the radius, through the relationship \(v = r \cdot \omega\).

Radius Calculation

Calculating the radius in circular motion involves understanding the relationship between linear speed, angular speed, and the radius itself. The radius is crucial as it directly impacts both speeds. To find the radius when you have the linear speed and angular speed, use the rearranged formula:

• \(r = \frac{v}{\omega}\)

This formula highlights how radius \(r\) connects to linear speed \(v\) and angular speed \(\omega\).

In our given scenario, with a linear speed of 5 feet per second and an angular speed of \(\frac{2\pi}{3}\) radians per second, you calculate the radius as:

• \(r = \frac{5}{\frac{2\pi}{3}} = \frac{15}{2\pi}\; feet\)

The computed radius, \(\frac{15}{2\pi}\), tells you how far you are sitting from the center of the ride. Understanding this computation is critical as it brings full circle how each of these values interrelate in circular motion dynamics.