A computer hard disk 8.0 cm in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at 600 rad/s² for ½ s, then coasts at a steady angular velocity for another ½ s. Through how many revolutions has the disk turned?

Verified step by step guidance

1. Determine the angular displacement during the acceleration phase using the kinematic equation for rotational motion: $\theta \; =\; \omega$₀t + ½αt². Here, $\omega$₀ is the initial angular velocity (0 rad/s), $\alpha$ is the angular acceleration (600 rad/s²), and $t$ is the time (0.5 s).
2. Calculate the angular velocity at the end of the acceleration phase using the equation: $\omega \; =\; \omega$₀ + αt. This will give the constant angular velocity during the coasting phase.
3. Determine the angular displacement during the coasting phase using the equation: $\theta \; =\; \omega t$, where $\omega$ is the angular velocity calculated in the previous step and $t$ is the time (0.5 s).
4. Add the angular displacements from the acceleration phase and the coasting phase to find the total angular displacement in radians.
5. Convert the total angular displacement from radians to revolutions using the conversion factor: $1\; \backslash (\backslash text\{\; revolution\}\backslash )\; =\; 2\pi \; \backslash (\backslash text\{\; radians\}\backslash )$.

Key Concepts

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). In this scenario, the disk accelerates at 600 rad/s², meaning its angular velocity increases by 600 radians per second every second. Understanding this concept is crucial for calculating how much the disk's angular velocity changes during the acceleration phase.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a fixed point, measured in radians. It can be calculated using the formula θ = ω_initial * t + 0.5 * α * t², where θ is the angular displacement, ω_initial is the initial angular velocity, α is the angular acceleration, and t is the time. This concept is essential for determining how far the disk has turned during the acceleration phase.

Revolutions

A revolution is a complete turn around a circle, equivalent to an angular displacement of 2π radians. To find the number of revolutions the disk has completed, the total angular displacement in radians must be divided by 2π. This concept is important for converting the angular displacement calculated from the previous phases into a more intuitive measure of rotation.