A long string is wrapped around a 6.0 -cm-diameter cylinder,...

A long string is wrapped around a $6.0$ -cm-diameter cylinder, initially at rest, that is free to rotate on an axle. The string is then pulled with a constant acceleration of $1.5 \mathrm{m} / \mathrm{s}^{2}$ until $1.0 \mathrm{m}$ of string has been unwound. If the string unwinds without slipping, what is the cylinder's angular speed, in $\mathrm{rpm}$, at this time?

Key Concepts

Rolling Without Slipping

This concept refers to a condition in rotational motion where the object (such as a cylinder or wheel) rotates in such a way that the point of contact with the surface has zero relative velocity. This implies that the linear displacement and velocity of the object are directly related to its angular displacement and angular velocity by the relationship v = ?r, where r is the radius. It is crucial for linking translational and rotational motion.

Angular Kinematics

Angular kinematics involves the study of rotational motion with constant angular acceleration, using equations analogous to those of linear kinematics. One key equation is ?² = ??² + 2??, where ? is the final angular speed, ?? is the initial angular speed, ? is the angular acceleration, and ? is the angular displacement. This framework is essential for solving problems where an object rotates under a constant acceleration.

Relationship Between Linear and Angular Quantities

Under the condition of no slipping, the linear acceleration (a) of a point on the rim of a rotating object is directly proportional to its angular acceleration (?) via the equation a = ?r. Similarly, linear displacement (s) relates to angular displacement (?) by s = r?. These relationships enable one to convert between translational and rotational measures, which is key in solving problems where both are involved.

Conversion Between Angular Speed Units

Angular speed can be expressed in various units. Converting from radians per second to revolutions per minute (rpm) requires understanding that one revolution is 2? radians and one minute is 60 seconds. The conversion is performed by multiplying the angular speed in rad/s by 60/(2?), making this concept crucial for interpreting results in different unit systems.

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