Problem 110 - Precalculus

Question

Find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$\omega=\frac{5 \pi \mathrm{rad}}{16 \mathrm{sec}}, r=24 \mathrm{ft}$$

Step-by-Step Solution

Answer

The linear speed is \( \frac{15\pi}{2} \text{ ft/sec} \).

Step 1: Understanding The Problem

We need to find the linear speed of a point on a circle. We know the circle has radius \( r = 24 \text{ ft} \) and is rotating with an angular speed of \( \omega = \frac{5\pi \text{ rad}}{16 \text{ sec}} \).

Step 2: Identifying The Formula

The linear speed \( v \) of a point on the circumference of a circle is related to the angular speed \( \omega \) and the radius \( r \) by the formula: \( v = r \omega \).

Step 3: Substituting Known Values

Substitute the given values for \( r \) and \( \omega \) into the linear speed formula: \[ v = 24 \times \frac{5\pi}{16} \].

Step 4: Performing The Calculation

Calculate the linear speed: First multiply the radius and angular speed, \( 24 \times \frac{5\pi}{16} = \frac{120\pi}{16} \). Simplify the fraction to get \( \frac{15\pi}{2} \).

Step 5: Final Result

The linear speed \( v \) is \( \frac{15\pi}{2} \text{ ft/sec} \).

Key Concepts

Angular Speed

Angular speed is a measure of how fast an object rotates or revolves relative to another point, usually the center of a circle. It's typically described in units of radians per second. Angular speed is different from linear speed, which measures how fast a point travels along a path. Instead, angular speed focuses on the rotation of the entire object.

The formula to calculate angular speed, denoted by \( \omega \), is:

• \( \omega = \frac{\theta}{t} \)

Here, \( \theta \) is the angular displacement in radians, and \( t \) is the time taken for this displacement. Therefore, if the angular displacement of a circle is \( 5\pi \) radians in 16 seconds, the angular speed \( \omega \) is \( \frac{5\pi}{16} \text{ rad/sec} \).

Understanding angular speed is crucial for determining linear speed because it forms part of the equation \( v = r \omega \), where \( v \) is the linear speed, \( r \) is the radius, and \( \omega \) refers to angular speed.By knowing \( \omega \) and \( r \), you can easily find \( v \), the speed of a point on the circle's edge.

Circumference of a Circle

The circumference of a circle is the total distance around the circle. It's similar to the perimeter you find for polygons, but it's specifically for round shapes. It's important because it influences how far a point travels in one complete revolution along the circle's edge.

The formula for the circumference \( C \) of a circle is:

• \( C = 2\pi r \)

where \( r \) is the radius of the circle. This means the circumference is directly dependent on the radius. The larger the radius, the larger the circumference.

Knowing the circumference is helpful when calculating distances traveled over time. If you are aware of the angular speed, you can calculate how many times an object goes around the circle in a given time frame, helping to determine the linear speed.

Radius

The radius of a circle is the distance from the center of the circle to any point on its boundary. It is an essential part of any calculation that involves circular movement because it helps define the size of the circle.

The radius influences several key properties of a circle, such as:

• The circumference \( C = 2\pi r \)
• The area \( A = \pi r^2 \)

Among these, the circumference is particularly important when discussing linear speed because it describes the circle's boundary.

When calculating linear speed \( v \) using angular speed \( \omega \), the radius is part of the formula \( v = r \omega \). Here, the radius acts as a scaling factor that converts angular speed into a linear context. For example, with a radius of 24 feet and an angular speed of \( \frac{5\pi}{16} \) rad/sec, the linear speed is \( \frac{15\pi}{2} \) ft/sec, calculated by multiplying the radius used in converting the angular motion to linear motion, which enables more practical understanding of speeds