The motor of a fan turns a small wheel of radius rm= 2.00 cm . This wheel turns a belt, which i

step 1 So to solve this question, let's first visualize the scenario. So there is a motor which is conn

The motor of a fan turns a small wheel of radius rm= 2.00...

The motor of a fan turns a small wheel of radius $r_{\mathrm{m}}=$ $2.00 \mathrm{~cm} .$ This wheel turns a belt, which is attached to a wheel of radius $r_{f}=3.00 \mathrm{~cm}$ that is mounted to the axle of the fan blades. Measured from the center of this axle, the tip of the fan blades are at a distance $r_{\mathrm{b}}=15.0 \mathrm{~cm} .$ When the fan is in operation, the motor spins at an angular speed of $\omega=1200$. rpm. What is the tangential speed of the tips of the fan blades?

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Key Concepts

Belt Drive and Gear Ratio

In systems using belts and pulleys, the belt transfer relates the rotational speeds of different wheels. Since the belt maintains a constant tangential speed at the circumference of the connected wheels, the angular speeds are inversely proportional to their radii. Consequently, the gear ratio, given by the ratio of the motor wheel’s radius to the fan wheel’s radius, determines how the motor’s angular speed is modified before affecting the fan blades’ tip speed.

Angular Speed Conversion

Angular speed is often given in revolutions per minute (rpm), which must be converted to radians per second (rad/s) when using the formula for tangential speed. This conversion is crucial because one revolution is 2? radians, and 1 minute is 60 seconds; thus, the conversion factor bridges the measurement units for proper computation in problems involving rotational motion.

Tangential Speed

Tangential speed is the linear speed of a point located at a distance from the center of rotation and is calculated as the product of the angular speed and the radial distance (v = r × ?). This concept is essential in understanding how angular motion translates into linear motion at the edge of rotating objects, such as the tip of a fan blade.

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