Problem 16-3: Engineering Mechanics- Dynamics (Hibbeler 14th Edition)

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1. We start with the constant angular acceleration equation: omega equals omega-naught plus alpha times t. Plugging in omega-naught equals 12 radians per second, alpha equals 20 radians per second squared, and t equals 2 seconds, we get omega equals 12 plus 40, which equals 52 radians per second.
2. Here is the kinematics diagram for the disk. Point A sits on the rim at a radial distance of 0.5 meters from the center. As the disk spins with angular velocity omega and angular acceleration alpha, point A has a velocity tangent to the rim, a tangential acceleration component in the tangential direction due to alpha, and a normal acceleration component pointing inward toward the center due to omega squared.
3. Here is the kinematics diagram for the disk. Point A sits on the rim at a radial distance of 0.5 meters from the center. As the disk spins with angular velocity omega and angular acceleration alpha, point A has a velocity tangent to the rim, a tangential acceleration component in the tangential direction due to alpha, and a normal acceleration component pointing inward toward the center due to omega squared.
4. The speed of point A equals the angular velocity multiplied by the radius. With omega equal to 52 radians per second and r equal to 0.5 meters, the speed is 52 times 0.5, giving 26 meters per second.
5. The tangential component of acceleration equals alpha times r. With alpha equal to 20 radians per second squared and r equal to 0.5 meters, the tangential acceleration is 20 times 0.5, which equals 10 meters per second squared.
6. The normal component of acceleration equals omega squared times r. With omega equal to 52 radians per second and r equal to 0.5 meters, we compute 52 squared, which is 2704, then multiply by 0.5 to get 1352 meters per second squared. This component points inward toward the disk's center.
7. To summarize: at t equals 2 seconds, the angular velocity of the disk is 52 radians per second. Point A has a speed of 26 meters per second. Its tangential acceleration component is 10 meters per second squared, and its normal acceleration component directed toward the center is 1352 meters per second squared. The normal component vastly dominates because the angular velocity has grown large after 2 seconds of constant angular acceleration.