An object is placed on an inclined plane. Does it roll?

Question

An object is placed on in inclined plane. There may or may not be friction, your choice. My question is, how do we figure out whether or not it rolls? For example a sphere rolls but a cube doesn't.

Answer 1

Gravity pulls in the center of gravity. It will be in the middle of both a cube and a ball in question.

There will be a normal force on the cube on the ends. There will be a normal force on the ball at the contact point.

If the weight pulls in the center further to the left (for a left directed incline) than where the normal force pushes, then the object will roll. The weight will then cause a torque to start the rotation. This is the case for the ball.

If the weight pulls in the center further to the right than the normal force, then the cause counteracting torques and no rotation will happen. This is the case for the cube, where the normal force will be at the left-most corner. To tilt the cube over, you must tilt it with your hands until the center of gravity is to the left of the corner, and then it will fall over.

Answer 2

You'd have to check to see if the sum of the moments around the toe causes a rolling rotation about the toe. In the case of a cylinder, the toe is positioned in such a way that the moment about the toe always causes rolling.

In general, draw up a free body diagram that shows all the forces and moments, focusing on the moments about the toe. Include gravity, friction, and normal forces. Even this simple situation is more complex than most people would imagine.

Answer 3

Look at the forces -- and more specifically the torque.

The sphere has the force of gravity pulling it downward, and the inclined plane pushing it out. The inclined plane exerts a force on the sphere at its contact point perpendicular to the plane. The force of gravity has some component along the plane, and some component perpendicular to the plane. The forces perpendicular are balanced, but the plane simply cannot balance a force along the plane. More precisely, measure the torque relative to the point of contact. Since the force from the plane is at the point of contact, $\vec{\tau} = \vec{r} \times \vec{F} = 0$, because $\vec{r}=0$. Gravity, on the other hand, exerts a nonzero torque (relative to this point of contact). So the sphere gets a nonzero net torque, and it rolls.

For a cube, on the other hand, the plane can exert force at multiple points. In particular, that leading edge exerts a torque to balance the torque of gravity.

Of course, if you tip the plane far enough, the cube will actually tip over...