Newton's second law F=ma and torque do not agree?

[lillybeans]

Homework Statement

Suppose a sphere is rolling on the ground and there is friction. Friction is the only net force acting on it, and by Newton's law, it will accelerate the object in the direction of the force (which is backwards). But the object is actually accelerating forward because friction is generating a positive torque.

Some equations:

Acceleration of center of mass = radius * angular acceleration (pure rolling motion)
Torque = radius x Force
Torque = moment of inertia * angular acceleration

Solving for acceleration of center of mass, I get two different answers. It seems that torque takes into account of the fact that the moment of inertia of a sphere is 2/5 mr^2 but Newton's second law doesn't. How to account for the differences? Am I doing something wrong?

 

[jbsteele]

Homework EquationsAcceleration of center of mass = radius * angular accelerationTorque = radius x ForceTorque = moment of inertia * angular accelerationThe Attempt at a SolutionWhen solving for the acceleration of the center of mass, you need to take into account the fact that the moment of inertia of a sphere is 2/5 mr^2. Newton's second law does not take this into account, but torque does.The equation for acceleration of center of mass is:Acceleration of center of mass = radius * angular accelerationThis equation can be rearranged to solve for angular acceleration:Angular acceleration = acceleration of center of mass / radiusThe equation for torque is:Torque = radius x ForceThis equation can also be rearranged to solve for angular acceleration:Angular acceleration = Torque / Moment of InertiaSince the moment of inertia of a sphere is 2/5 mr^2, the equation for angular acceleration becomes:Angular acceleration = (radius x Force) / (2/5 mr^2)Using this equation, we can solve for the acceleration of the center of mass:Acceleration of center of mass = radius x (Force / (2/5 mr^2))Therefore, when solving for the acceleration of the center of mass, you need to take into account the moment of inertia of the sphere. Since Newton's second law does not take this into account, you cannot use it to solve for the acceleration of the center of mass. You must use torque instead.