Question

After you pick up a spare, your bowling ball rolls without slipping back toward the ball rack with a linear speed of \(2.8 \mathrm{~m} / \mathrm{s}\) as shown in Figure \(8.28\). (a) If the diameter of the bowling ball is \(0.22 \mathrm{~m}\), what is its angular speed? (b) To reach the rack, the ball rolls up a ramp. If the angular speed of the ball when it reaches the top of the ramp is \(1.2 \mathrm{rad} / \mathrm{s}\), what is the linear speed of the ball?

Short answer

(a) Angular speed is approximately 25.45 rad/s. (b) Linear speed is approximately 0.132 m/s.

Step-by-step solution

Understand Angular Speed

The angular speed \( \omega \) can be found using the relation between linear speed \( v \) and angular speed: \( \omega = \frac{v}{r} \), where \( r \) is the radius of the ball.

Calculate the Angular Speed for Part (a)

Given that the diameter of the ball is \(0.22\,m\), the radius \( r \) is \( \frac{0.22}{2}\,m = 0.11 \, m \). The linear speed \( v \) is \( 2.8 \, m/s \). Therefore, the angular speed is \( \omega = \frac{2.8}{0.11} \, \mathrm{rad/s} \approx 25.45 \, \mathrm{rad/s} \).

Understand Linear Speed from Angular Speed

For part (b), we have the angular speed \( \omega \) given as \( 1.2\, \mathrm{rad/s} \). We need to find the linear speed \( v \) using the same formula rearranged: \( v = \omega \times r \).

Calculate the Linear Speed for Part (b)

Using the radius \( r = 0.11 \, m \) and the angular speed \( \omega = 1.2 \, \mathrm{rad/s} \), the linear speed \( v = 1.2 \times 0.11 \approx 0.132 \, \mathrm{m/s} \).

Key concepts

Angular Speed

Angular speed is a measure of how fast an object rotates or spins. Specifically, it refers to the angle an object turns through per unit of time.
For a rotating object, like a bowling ball, this is an important concept. To calculate angular speed (\( \omega \)), we use the formula: \[ \omega = \frac{v}{r} \]Here

• \( \omega \) is the angular speed in radians per second (rad/s)
• \( v \) is the linear speed of the object (m/s)
• \( r \) is the radius of the circle the object makes (m)

This relation tells us that the farther an object is from the center of rotation, the smaller its angular speed for the same linear speed.
This formula allows us to see how changes in linear speed or the radius affect the rotation rate.

Linear Speed

Linear speed refers to the rate at which an object moves along a path. It is the speed you typically think of when imagining objects in motion, like cars on a street or a rolling bowling ball. In physics, this is often denoted by \( v \) and measured in meters per second (m/s).
The relationship between linear and angular speed is crucial when dealing with rotational motion. Using the formula \[ v = \omega \times r \]we can determine the linear speed once we know the angular speed \( \omega \) and the radius \( r \). So, if you increase the angular speed or the radius, the linear speed will also increase as a direct result.
This concept helps us understand behaviors like why larger wheels allow vehicles to travel faster at the same rotational speed.

Moment of Inertia

The moment of inertia is a key concept in physics, especially when dealing with rotational motion. It represents the distribution of mass in an object and determines how difficult it is to change the object's rotational speed.
Think of it as the rotational equivalent of mass for linear motion. The moment of inertia depends on:

• Mass of the object (the heavier the object, the greater the moment of inertia)
• Distribution of that mass relative to the axis of rotation (mass further from the axis increases the moment of inertia)

For simple shapes, there are standard formulas to calculate this value. Understanding the moment of inertia is crucial because it helps predict how objects will behave when forces are applied. It's what ballet dancers use to spin faster by pulling in their arms, reducing their moment of inertia.
In our bowling ball example, a low moment of inertia would mean it accelerates more easily when starting to roll.

Physics Education

Physics education involves learning how fundamental principles explain the world around us. It's not just about memorizing formulas; it's about understanding how and why formulas work.
This way of thinking can be applied to various scenarios, like the exercise involving a bowling ball. By learning about angular and linear speeds, and moments of inertia, students gain insights into:

• How everyday objects move and behave, fostering a deeper connection with the physical world
• Problem-solving skills by applying theoretical knowledge to real-world situations
• Critical thinking by understanding the interrelationship of different physical concepts

Teaching these concepts with practical examples makes them more relatable and easier to grasp. Good physics education emphasizes understanding over rote learning, helping students become proficient problem solvers.
This kind of education supports the development of skills that are extremely valuable, not just in physics, but in life.