Final answer:
To find the angular acceleration of a disk and a hoop rolling down an incline, one must calculate the linear acceleration using Newton's second law and then divide the result by the radius. The moment of inertia plays a crucial role, as it is different for a disk and a hoop, leading to different angular accelerations for each.
Step-by-step explanation:
The question involves a disk and a hoop rolling down a 30-degree incline. Both have the same mass and radius and start from rest. To find the angular acceleration, we can use the formula:
α = a/r,
where α is the angular acceleration, a is the linear acceleration, and r is the radius. Since they roll without slipping, the linear acceleration a can be found using Newton's second law and considering rotational motion. For a disk, the moment of inertia I is (1/2)MR², and for a hoop, it is MR².
The free body diagram for both the disk and hoop will have gravitational force components and no frictional force components, assuming that both are rolling without slipping. This will lead to the equation:
mg sin(θ) = (I + MR²) * a/R,
where θ is the angle of the incline, M is the mass of the disk or hoop, and g is the acceleration due to gravity (9.8 m/s²). Plugging in the known values and solving for a, we can then calculate α.
The calculations will show that the angular acceleration is different for the disk and hoop due to their different moments of inertia.