Problem 25 - Applied Physics

Question

A wheel of radius \(27.0 \mathrm{~cm}\) has an angular speed of \(47.0 \mathrm{rpm}\). Find the linear speed (in \(\mathrm{m} / \mathrm{s}\) ) of a point on its rim.

Step-by-Step Solution

Answer

The linear speed is approximately 1.33 meters per second.

Step 1: Convert RPM to Radians per Second

The angular speed is given as 47 rotations per minute (rpm). First, convert this to radians per second. There are \(2\pi\) radians in a full rotation, and 60 seconds in a minute. So, \(47 \mathrm{~rpm} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{47 \times 2 \pi}{60}\).

Step 2: Calculate Angular Speed in Radians per Second

Calculate the above expression: \(\frac{47 \times 2 \pi}{60} = \frac{94 \pi}{60}\) radians per second. Simplifying gives us approximately \(4.93\) radians per second.

Step 3: Determine Linear Speed

The linear speed \(v\) is the product of the angular speed \(\omega\) in radians per second, and the radius \(r\) in meters: \(v = \omega \times r\). Here, \(r = 27.0 \text{ cm} = 0.27 \text{ m}\).

Step 4: Calculate Linear Speed

Using the formula \(v = \omega \times r\), substitute \(\omega = 4.93\) radians/second and \(r = 0.27 \text{ m}\):\(v = 4.93 \times 0.27 \). Calculate this to get \(v \approx 1.33 \text{ m/s}\).

Key Concepts

Linear Speed

Linear speed refers to the distance traveled by a point moving along a straight path in a given time. For instance, when you consider a point on the rim of a rotating wheel, this point moves linearly, tracing a circular path.

The linear speed (\( v \) ) of this point is linked to the wheel's angular speed (\( \omega \) ) and the radius (\( r \) ) of the wheel. The formula to calculate linear speed is:

• \( v = \omega \times r \)

To convert the radius from centimeters to meters, remember that \( 1 ext{ cm} = 0.01 ext{ m} \) . So, a radius of \( 27.0 ext{ cm} \) becomes \( 0.27 ext{ m} \) . Once you have the angular speed in radians per second, you multiply it by the radius in meters to get the linear speed in meters per second. This provides a clear understanding of how fast a point on the wheel's rim is moving.

Radians per Second

When we talk about angular speed, we often use the unit 'radians per second' (\( ext{rad/s} \) ). But what is a radian? Radians measure angles via the radius of a circle, with \( 2\pi \) radians in a full circle. Instead of degrees, which break down a circle into 360 parts, radians divide it using pi. This is why converting RPM (rotations per minute) into \( ext{rad/s} \) is essential in our calculations.

Given \( 47 ext{ rpm} \) , we first note that each rotation equals \( 2\pi \) radians. Then, we convert the time from minutes to seconds:

• \( 47 \times \frac{2\pi}{60} \approx 4.93 \text{ rad/s} \)

With this conversion, now you can use the angular speed to calculate linear speed. This unit change is crucial for calculations involving physics and engineering matters.

Conversion of Units

Converting units is foundational in physics. When working with rotational speeds, being able to shift between units like RPM, radians, and meters per second is crucial.

For the given problem:

• The first conversion was RPM to radians per second to obtain angular speed.
• The second conversion involved changing the radius from centimeters to meters.

It is crucial because meters are the standard unit for measuring length in the International System of Units (SI), which makes calculations clear and consistent.
For instance, \( 60 \text{ seconds} \) makes \( 1 ext{ minute} \) , and \( 100 ext{ cm} \) equals \( 1 ext{ m} \) . These are basic unit conversion factors essential for making sure everyone gets the same answer, no matter where they are or the unit system they usually use.

Wheel Radius

The radius of a wheel is simply half the diameter of the wheel. It's a crucial factor in determining the linear speed of a point on the wheel.

Why the radius? Because in the formula for linear speed, \( v = \omega \times r \) , the radius directly scales the angular speed to produce a linear speed. A larger radius spins a larger path, and thus results in a faster linear speed for the same angular speed.

For conversion, remember: \( 27 ext{ cm} \) means \( 0.27 ext{ m} \) . This conversion is important for performing calculations in a standardized metric system. Understanding the role of the radius not only helps solve the given problem but also aids in visualizing how changes in radius could affect the motion characteristics of any rotating object. Additionally, this understanding is vital in applications involving gears, pulleys, and other rotating systems.