Many 2.5-in-diameter (6.35-cm) computer hard disks spin at a...

Many 2.5-in-diameter (6.35-cm) computer hard disks spin at a constant 7200 rpm operating speed. The disks have a mass of about $7.50 \mathrm{~g}$ and are essentially uniform throughout with a very small hole at the center. If they reach their operating speed $2.50 \mathrm{~s}$ after being turned on, what average torque does the disk drive supply to the disk during the acceleration? Example $8-12$

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear dynamics. For a uniform disk, the moment of inertia can be calculated with the formula I = 0.5MR². It plays a critical role in determining how the applied torque affects the angular acceleration of the body.

Angular Velocity Conversion

Angular velocity conversion involves converting rotational speeds from one unit, such as revolutions per minute (rpm), to radians per second (rad/s). This conversion is essential because it allows the use of standard SI units for consistency in rotational dynamics calculations, where 1 revolution equals 2? radians and 1 minute equals 60 seconds.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It quantifies how quickly an object speeds up or slows down its rotation. In problems involving a constant acceleration, it is determined by dividing the change in angular velocity by the time interval over which the change occurs.

Torque

Torque is the measure of the force that causes an object to rotate about an axis. It is directly related to the angular acceleration of an object and is computed using Newton's second law for rotational motion, which states that torque equals the moment of inertia times the angular acceleration (? = I?). This concept is fundamental when analyzing rotational acceleration in mechanical systems.

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