Students will learn how to solve problems of rolling objects and also of objects that are rolling and sliding.

### Key Equations

When an object is rolling without slipping this means that and . This is also true in the situation of a rope on a pulley that is rotating the pulley without slipping. Using this correspondence between linear and angular speed and acceleration is very useful for solving problems, but is only true if there is no slipping. Also, know that when the object is sliding, kinetic friction is in play. When it is rolling, then static friction. Often, an object will start out rolling and sliding (kinetic friction) until it slows enough that it is rolling without sliding (static friction). One can set up a condition by forcing and in order to find the point where it stops sliding. Finally, when this rolling object rolls down the incline, it gains kinetic energy and loses potential energy, just like any object going down an incline. However, for rolling objects the kinetic energy is split between two forms: rotational and kinetic:

Remember, if the object isn't sliding but perfectly rolling:

#### Example 1

You throw a bowling ball of mass and radius with an initial speed down a flat bowling lane with a coefficient of kinetic friction . Initially, the ball slides down the lane not rotating at all, but after a time , it begins to roll perfectly without sliding. Find in terms of the values given above.

##### Solution

We'll start by drawing an FBD for the bowling ball.

We'll start by applying Newton's second law to the bowling ball. The force of friction is the only force in the x-direction.

We now have two unknowns in our equation so we'll use Newton's second law for rotation as our second equation to help us solve this problem. Using Newton's second law for rotation, we're going to determine a value for in terms of the other values.

Now, we can put that value we just found back into our Newton's second law equation and solve for .

### Watch this Explanation

### Time for Practice

- A solid cylinder of mass, M, and radius R rolls without slipping down an inclined plane that makes an angle . The cylinder starts from rest at a height H. The inclined plane makes an angle with the horizontal. Express all solutions in terms of M, , R, H, and g.
- Draw the free body diagram for the cylinder.
- Determine the acceleration of the center of mass of the cylinder while it is rolling down the inclined plane.
- Determine the minimum coefficient of friction between the cylinder and the inclined plane that is required for the cylinder to roll without slipping.
- The coefficient of friction μ is now made less than the value determined in part d so that the cylinder both rotates and slips. How does the translational speed change from above (i.e. larger, smaller, same). Justify your answer.

- For a ball rolling without slipping with a radius of , a moment of inertia of , and a linear velocity of calculate the following:

#### Solutions

- a = 2gsinθ/3 b. (tan θ) /3 c. The translational, or linear speed increases because some of the energy that would have gone into rotational kinetic energy now goes to linear kinetic energy, hence making the linear speed greater.
- a. b. c. d.