Final answer:
To calculate the angular acceleration of the wheel down the incline, apply Newton's second law for rotation using the torque provided by friction, the wheel's moment of inertia based on its radius of gyration, and the weight of the wheel. Substitute the given values into the equations and solve for angular acceleration.
Step-by-step explanation:
To determine the wheel's angular acceleration as it rolls down the incline without slipping, we can use Newton's second law for rotation. The equation for rotational dynamics is τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. For a wheel rolling without slipping, the friction force provides the torque, and so the torque can be calculated as the product of the force of gravity component along the incline (mg sinθ), the wheel's radius (r), and the cosine of the angle to account for the perpendicular distance (τ = r × F × cosθ). The force can be found using F = mg sinθ, where m is the mass, g is the acceleration due to gravity, and θ is the angle of the incline.
The moment of inertia for a wheel can be calculated using the radius of gyration (kₗ), given by I = m × kₗ². With this information, we can find the angular acceleration (α) by isolating α in the equation τ = Iα, giving us α = τ / I. In this case, the weight (W) is provided instead of mass, where W = mg. Replacing mg with W in our force equation and calculating for α using the given values for W, r, kₗ, and θ will provide the answer to the question.