Rotation without slipping

Question

A solid cylinder rolls without slipping on an incline of angle θ. Calculate its linear acceleration downhill.

1. Isn't this a rotational motion? Why am I calculating a linear acceleration?
2. What does "roll without slipping" mean in physics terms?

Answer 1

Rolling w/o slipping is an often-tested two-motion problem, it involves both a linear motion and an angular (rotational) motion. Physically, it is a constraint that always keeps linear acceleration a = r * alpha as John-Paul wrote. If this constraint is not satisfied, imagining a car slipping in the mud, the car won't be able to move forward even though the tire rotates fiercely.

The other point tested here is how to calculate angular acceleration alpha based on the external torque and the moment of inertia of the object. The torque here is provided by the gravitational force, the center of rotation is the contact point (not the center!). So you need to apply the parallel axis theorem to get the right moment of inertia.

Answer 2

I won't show the calculation but instead simply answer your two questions and hopefully that is enough for you to calculate the acceleration yourself.

The cylinder is rolling down the incline but if you consider the center of gravity (the middle of the cylinder) alone you can think of it as a particle sliding down the hill. The part that may be confusing is that the cylinder does have an angular (rotational) acceleration as well that is directly proportional to the linear acceleration. So you should actually be able to find both!

This brings me to your next question. Rolling without slipping in physics means that the cylinder won't ever slide down the hill, it will only be allowed to roll. What this means is the distance the cylinder has rolled down the incline and the amount the cylinder has rotated are directly proportional. As far as equations go, this is were the linear acceleration and angular acceleration are related by a=r*(alpha). Where "a" is the linear acceleration, "r" is the radius of the cylinder, and "alpha" is the angular acceleration.