A cylinder rolls down an incline without slipping. If its mass is M and radius i

A cylinder rolls down an incline without slipping. If its mass is M and radius is R, what is the acceleration of its center of mass? (2020)

A cylinder rolls down an incline without slipping. If its mass is M and radius is R, what is the acceleration of its center of mass? (2020)

Step 1: Understand that the cylinder is rolling down an incline, which means it is both translating (moving down) and rotating (spinning).
Step 2: Identify the forces acting on the cylinder. The main force is gravity, which can be broken down into two components: one acting down the incline (M * g * sin(θ)) and one acting perpendicular to the incline.
Step 3: Recognize that for rolling without slipping, the linear acceleration of the center of mass (a_cm) is related to the angular acceleration (α) by the equation a_cm = R * α.
Step 4: Use Newton's second law for rotation. The torque (τ) caused by the gravitational force down the incline is τ = R * (M * g * sin(θ)). This torque causes the cylinder to rotate.
Step 5: Relate torque to angular acceleration using τ = I * α, where I is the moment of inertia of the cylinder. For a solid cylinder, I = (1/2) * M * R^2.
Step 6: Set up the equation: R * (M * g * sin(θ)) = (1/2) * M * R^2 * α. Substitute α with a_cm / R to get R * (M * g * sin(θ)) = (1/2) * M * R^2 * (a_cm / R).
Step 7: Simplify the equation to find a_cm. Cancel R from both sides and rearrange to get a_cm = (2/3) * g * sin(θ).
Step 8: Conclude that the acceleration of the center of mass of the cylinder rolling down the incline is a_cm = (2/3) * g * sin(θ).

Rolling Motion – Understanding the dynamics of a rolling object, including the relationship between translational and rotational motion.
Inclined Plane – Analyzing forces acting on an object on an incline, including gravitational force components.
Moment of Inertia – Applying the moment of inertia for a solid cylinder to derive the acceleration of the center of mass.