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 umwound. If the string unwinds without slipping, what is the cylinder's angular speed, in rpm, at this time?

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Key Concepts

Unit Conversion between Radians per Second and revolutions per Minute

Angular speed is often calculated in radians per second using rotational kinematics, but it can be more intuitively understood in terms of revolutions per minute (rpm). Converting between the two units involves recognizing that one full revolution is 2? radians and that there are 60 seconds in a minute, so multiplying the angular speed in rad/s by 60/(2?) yields the speed in rpm.

Linear to Angular Relationship (Non?Slip Condition)

When a string unwinds without slipping from a cylinder, the distance the string moves corresponds directly to the arc length traversed by the surface of the cylinder. This direct relationship means that the linear acceleration (and displacement) experienced by the string is directly connected to the angular acceleration (and angular displacement) of the cylinder, using the relation a = r? and s = r?.

Rotational Kinematics

Rotational kinematics uses equations analogous to those in linear motion to analyze quantities like angular displacement, angular velocity, and angular acceleration when an object rotates. In problems with constant angular acceleration, formulas similar to v² = v?² + 2a?s are applied in the form ?² = ??² + 2??, enabling determination of angular speed after a given rotation.

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