Angular Speed of a Pulley. Two pulleys, 50 cm and 30 cm in diameter, respectively, are connected

step 1 The first thing we do is recognize the fact that when two pulleys are attached together, their v

Angular Speed of a Pulley. Two pulleys, 50 cm and 30 cm in...

Angular Speed of a Pulley. Two pulleys, $50 \mathrm{cm}$ and $30 \mathrm{cm}$ in diameter, respectively, are connected by a belt. The larger pulley makes 12 revolutions per minute. Find the angular speed of the smaller pulley, in radians per second.
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Key Concepts

Belt Drive Systems

Belt drive systems connect two or more pulleys such that the belt transmits motion between them. A key principle in these systems is that the linear speed at the rim of each pulley is constant along the belt, regardless of the pulley's radius. This invariant belt speed facilitates the computation of unknown angular speeds when the sizes and rotational speeds of the pulleys are given.

Relationship between Linear and Angular Speed

The connection between linear and angular speed is defined by the equation v = ?r, where v is the linear speed, ? is the angular speed, and r is the radius. This relationship is crucial in problems where the motion of a belt or rim attached to a rotating object must be analyzed, allowing one to convert between the linear speed of the belt and the angular speed of the pulleys.

Conversion between Revolutions per Minute and Radians per Second

Converting between revolutions per minute (rpm) and radians per second is necessary when solving rotational motion problems. Since one full revolution corresponds to 2? radians and time conversion factors connect minutes and seconds, this conversion ensures that angular speed is expressed in standard SI units, facilitating calculations and comparisons.

Angular Speed

Angular speed measures how fast an object rotates, typically expressed in radians per second. It quantifies the rate of change of the rotational angle and is essential in understanding rotational motion in systems like pulleys or wheels.

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