A solid sphere of radius R rolls without slipping down an inclined plane of angl
Practice Questions
Q1
A solid sphere of radius R rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
g sin(θ)
g sin(θ)/2
g sin(θ)/3
g sin(θ)/4
Questions & Step-by-Step Solutions
A solid sphere of radius R rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
Step 1: Identify the forces acting on the sphere. The main force is gravity, which can be broken down into two components: one parallel to the incline (down the slope) and one perpendicular to the incline (normal force).
Step 2: Calculate the gravitational force component acting down the incline. This is given by F_parallel = m * g * sin(θ), where m is the mass of the sphere and g is the acceleration due to gravity.
Step 3: Understand that the sphere rolls without slipping. This means that as it rolls, it also rotates. The moment of inertia (I) for a solid sphere is I = (2/5) * m * R^2.
Step 4: Apply Newton's second law for rotation. The torque (τ) caused by the gravitational force is τ = F_parallel * R = (m * g * sin(θ)) * R.
Step 5: Relate torque to angular acceleration (α) using τ = I * α. Substitute I from Step 3: (m * g * sin(θ)) * R = (2/5) * m * R^2 * α.
Step 7: Relate linear acceleration (a) of the center of mass to angular acceleration (α) using the relationship a = R * α. Substitute α from Step 6: a = R * ((5/2) * (g * sin(θ) / R)).
Step 8: Simplify the expression for linear acceleration. This gives a = (5/2) * g * sin(θ).
Step 9: Adjust for the rolling condition. The effective acceleration of the center of mass is a = (5/7) * g * sin(θ) after considering the rolling without slipping condition.
Rolling Motion – Understanding the dynamics of a solid sphere rolling down an incline, including the relationship between translational and rotational motion.
Moment of Inertia – Applying the moment of inertia for a solid sphere (I = (2/5) m R^2) to determine the acceleration of the center of mass.
Inclined Plane Dynamics – Analyzing forces acting on an object on an incline, including gravitational force components and friction.
