A cylinder rolls down an incline of angle θ. What is the acceleration of the cen

₹0.0

Question: A cylinder rolls down an incline of angle θ. What is the acceleration of the center of mass of the cylinder?

Options:

Correct Answer: g sin(θ)/2

Solution:

The acceleration a = g sin(θ) / (1 + k^2/r^2) where k^2/r^2 for a solid cylinder is 1/2. Thus, a = g sin(θ) / (1 + 1/2) = g sin(θ)/2.

A cylinder rolls down an incline of angle θ. What is the acceleration of the center of mass of the cylinder?

A cylinder rolls down an incline of angle θ. What is the acceleration of the center of mass of the cylinder?

Step 1: Identify the forces acting on the cylinder as it rolls down the incline. The main force is gravity, which can be broken down into two components: one parallel to the incline (g sin(θ)) and one perpendicular to the incline.
Step 2: Understand that the cylinder rolls without slipping, which means it has both translational and rotational motion.
Step 3: Recall the formula for the acceleration of the center of mass of a rolling object: a = g sin(θ) / (1 + k^2/r^2), where k is the radius of gyration and r is the radius of the cylinder.
Step 4: For a solid cylinder, the value of k^2/r^2 is 1/2. Substitute this value into the formula.
Step 5: Simplify the equation: a = g sin(θ) / (1 + 1/2) = g sin(θ) / (3/2).
Step 6: Further simplify to find the final acceleration: a = (2/3) g sin(θ).

Rolling Motion – Understanding the dynamics of rolling objects, including the relationship between translational and rotational motion.
Moment of Inertia – Applying the concept of moment of inertia (k^2/r^2) to determine the acceleration of the center of mass for different shapes.
Inclined Plane Dynamics – Analyzing forces acting on an object on an incline, including gravitational force components.