We now consider the combination of rolling motion and translational motion. Two assumptions are made when we consider rolling motion. One is that the axis of rotation passes through the center of mass and the other is that the axis of rotation always points in the same direction. When an object is rolling it has both linear kinetic energy and rotational kinetic energy. Thus the total kinetic energy of a rolling object is the sum of the two kinetic energies. Note that the linear velocity is the center of mass velocity. For rolling motion without slipping there is an important equation that relates angular motion with linear motion. This relationship states that the velocity of the center of mass is equal to the radius of the object times the angular velocity. Equivalently, the acceleration of the center of mass is equal to the radius of the object times the angular acceleration. These equations are sometimes referred to as the rolling without slipping conditions. We now consider a solid sphere rolling down an inclined plane. Lets see if we can calculate the acceleration of the sphere as it rolls down. Note that the friction between the sphere and the surface is what causes the ball to rotate. Without friction the ball would simply slide down without rolling. Since we are looking for the acceleration, the best place to start is usually Newton's second law. Taking x to be positive in the direction which the sphere is rolling, what is the linear force equation for the x component? ... What is the torque equation? ... We have two equations, but we have three unknowns, the friction, the acceleration and the angular acceleration. We can also use the rolling without slipping condition that states that the acceleration is equal to the radius times the angular acceleration. Now, given the moment of inertia of a sphere, what is the acceleration? The factor of five-sevenths in the acceleration is due to the moment of inertia of the sphere. If the ball were to slide down the incline without rolling the acceleration would be g times sine theta and thus it would reach the bottom faster. This is because friction, which causes the sphere to roll, acts against motion and decreases the linear acceleration. The acceleration of an object rolling down an incline is dependent on the moment of inertia of the object. Objects with a smaller moment of inertia will have a larger acceleration. If the three objects at the top of the incline all have the same mass, which one would roll to the bottom first? We see that the solid sphere, which has the smallest moment of inertia of the three objects, reaches the bottom first. This result can also be found using energy considerations. Recall that the total kinetic energy of a rolling object is the sum of the linear and rotational kinetic energies. We know from conservation of energy that when the objects reach the bottom of the incline, the total kinetic energy will be equal to the gravitational potential energy, which is the mass times gravity times the height of the incline. Now use the rolling without slipping condition in the conservation of energy expression. From this expression you can solve for the velocity of the center of mass when the objects hit the bottom of the incline. Notice that when the moment of inertia is larger the final velocity is smaller, and thus the object takes longer to reach the bottom. This is because some of the energy is converted into rotational motion and an object with a larger moment of inertia will have more rotation.

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