1. **State the problem:** We need to find the linear speed of a belt in centimeters per second, expressed exactly in terms of $\pi$. 2. **Understand the relationship:** The linear speed $v$ of a belt moving around a circular pulley is related to the angular speed $\omega$ (in radians per second) and the radius $r$ of the pulley by the formula: $$v = r \times \omega$$ 3. **Important rules:** - Angular speed $\omega$ is often given in revolutions per second (rev/s). To convert to radians per second, multiply by $2\pi$ because one revolution equals $2\pi$ radians. - Radius $r$ must be in centimeters to get linear speed in cm/s. 4. **Apply the formula:** - Suppose the pulley has radius $r$ cm and rotates at $f$ revolutions per second. - Then angular speed is $\omega = 2\pi f$ radians per second. - Linear speed is: $$v = r \times 2\pi f = 2\pi r f$$ 5. **Final answer:** The linear speed of the belt is exactly $$\boxed{2\pi r f \text{ cm/s}}$$ where $r$ is the radius in centimeters and $f$ is the rotational frequency in revolutions per second. If you provide the values of $r$ and $f$, I can compute the exact linear speed for you.