Suppose that a belt drives two wheels of radii rand R, as...

Suppose that a belt drives two wheels of radii rand $R,$ as indicated in the figure. (FIGURE CAN'T COPY) If $r=6 \mathrm{cm}, R=10 \mathrm{cm},$ and the angular speed of the larger wheel is 100 rpm, determine each of the following: (a) the angular speed of the larger wheel in radians per minute; (b) the linear speed of a point on the circumference of the larger wheel; (c) the angular speed of the smaller wheel in radians per minute. Hint: Because of the belt, the linear speed of a point on the circumference of the larger wheel is equal to the linear speed of a point on the circumference of the smaller wheel. (d) The angular speed of the smaller wheel in rpm.

Suppose that a belt drives two wheels of radii rand $R,$ as indicated in the figure.
(FIGURE CAN'T COPY)
If $r=6 \mathrm{cm}, R=10 \mathrm{cm},$ and the angular speed of the larger wheel is 100 rpm, determine each of the following:
(a) the angular speed of the larger wheel in radians per minute;
(b) the linear speed of a point on the circumference of the larger wheel;
(c) the angular speed of the smaller wheel in radians per minute. Hint: Because of the belt, the linear speed of a point on the circumference of the larger wheel is equal to the linear speed of a point on the circumference of the smaller wheel.
(d) The angular speed of the smaller wheel in rpm.

Key Concepts

Angular Speed Conversion

Angular speed can be expressed in different units, such as revolutions per minute (rpm) and radians per minute. Converting between these units is important because many formulas in rotational dynamics require angular speed to be in radians. The conversion factor used is that one complete revolution is equal to 2? radians.

Linear Speed Calculation

The linear speed (or tangential speed) of a point on the circumference of a rotating wheel is determined by the product of the wheel's radius and its angular speed (measured in radians per unit time). This relationship, often represented by the equation v = r × ?, is fundamental in connecting rotational movement with linear motion.

Belt Drive Principle

In belt-driven systems, the belt imposes a constraint whereby the linear (tangential) speeds at the rims of the connected wheels are the same. This principle allows us to relate the angular speeds of the two wheels, even though their radii might differ.

Relationship between Radii and Angular Speeds

Given that the linear speeds of the two wheels are equal in a belt drive system, their angular speeds are inversely proportional to their radii. Mathematically, if one wheel has radius R and angular speed ??, and the other has radius r and angular speed ??, then R × ?? = r × ??. This relationship is crucial when solving for unknown rotational speeds in mechanical systems involving belts.

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