Question

A bicycle wheel with a radius of \(0.62 \mathrm{~m}\) rotates with an angular speed of \(21 \mathrm{rad} / \mathrm{s}\) about its axle, which is at rest. What is the linear speed of a point on the rim of the wheel?

Short answer

The linear speed is \( 13.02 \) m/s.

Step-by-step solution

Identify Known Values

We are given the radius of the wheel as \( r = 0.62 \) meters and the angular speed \( \omega = 21 \) radians per second.

Use the Relationship Between Linear and Angular Speed

The formula to relate linear speed \( v \) and angular speed \( \omega \) is \( v = r \cdot \omega \), where \( r \) is the radius of the wheel and \( \omega \) is the angular speed.

Substitute Values into the Formula

Substitute the given values into the formula: \( v = 0.62 \times 21 \).

Calculate the Linear Speed

Perform the multiplication: \( 0.62 \times 21 = 13.02 \).

State the Final Answer

The linear speed \( v \) of a point on the rim of the wheel is \( 13.02 \) meters per second.

Key concepts

angular speed

Angular speed is a measure of how quickly something rotates or spins about an axis. It tells us the change in the rotational angle per unit time. This is usually denoted by the Greek letter \(\omega\). Angular speed is normally measured in radians per second (\text{rad/s}). The radian is a special unit of angular measurement where the angle is considered in terms of the radius. One full rotation around a circle is equal to \(2\pi\) radians, akin to 360 degrees in regular angles.
Angular speed is pivotal in understanding circular motion, as it helps demonstrate how quickly a point on a rotating object moves around the circle. For instance, if a bicycle wheel rotates with an angular speed of 21 \text{rad/s}, that signifies that any point on the wheel will execute a rotational movement of 21 radians every second.
In summary:

• Angular speed is how fast something spins around an axis.
• Measured in radians per second.
• Pivotal for describing circular or rotational motion.

bicycle wheel

The bicycle wheel is a perfect real-world example to understand angular and linear speeds. It's a cylindrical object rotating around an axis. The tire or rim of the wheel is what interacts with the ground when the bicycle moves, and it demonstrates the fundamental principles of rotational motion.
When a bicycle pedal is turned, it causes the chain to move, which in turn rotates the rear bicycle wheel. The faster you pedal, the faster the rotation, increasing both angular speed and the resulting linear speed. Understanding a bicycle wheel's properties, such as radius and angular speed, can help determine how fast you are traveling.
Key Points:

• Bicycle wheels rotate, making them ideal examples for studying rotational motion.
• The rotation of the wheel directly impacts the bike's speed on the road.
• Changing either the angular speed or wheel size affects the ride speed.

radius

The radius is the distance from the center of a circle to any point on its circumference. In rotation-related problems like those involving a bicycle wheel, the radius is crucial for determining linear speed. The formula \(v = r \cdot \omega\) highlights the importance of the radius, where \(v\) is linear speed, \(r\) is radius, and \(\omega\) is angular speed.
A larger radius implies the point on the wheel covers more distance as it completes a rotation, thus impacting the linear speed significantly. For instance, in a bicycle wheel with a radius of 0.62 meters, each complete rotation correlates to a greater travel distance on the ground.
Remember:

• The radius is the measure from the center to the edge of a circle.
• Plays a critical role in defining linear speed in rotational dynamics.
• Changing the radius will affect how far each rotation travels.