Ball rolling down a surface - where is the axis of rotation?

• lillybeans
In summary, the textbook explains that the point of contact between a rolling ball and a surface can be considered the pivot or axis of rotation, with zero velocity at that point. However, in problem solutions, the point of contact is redefined as the center of mass, leading to discrepancies in the calculation of torque. This is because gravity always passes through the axis of rotation, but friction is affected by its location. The choice of axis of rotation and center of mass is a matter of defining a frame of reference and does not change the physical behavior of the system.

lillybeans

In my textbook, it says that when a ball rolls down a surface, inclined or not, the point of contact can be thought of as the pivot or the axis of rotation, since at the point of contact, the ball's velocity is zero. Makes sense, but then the textbook becomes inconsistent in problems solving.

There are a few exercises of spheres rolling down surfaces, both inclined and non-inclined. If the point of contact between the ball and the surface is the axis of rotation, then the torque done by friction would be zero because it passes through the axis of rotation! In other words, it acts AT the pivot, the point of contact. But in the solutions, they clearly say the torque done by friction is "fR, where R is the radius of the ball". But aren't they now REDEFINING the axis to be the center of mass, hence the "R"?

I guess gravity is not affected by the axis of rotation in either of these cases cause it always passes through the axes of rotation in both cases. But friction is.

Can anyone explain the discrepancy? Thank you in advance!

The point of contact is called the instantaneous center. The torque is not zero because the force (mg) does not pass through it.

Axis of rotation is wherever you define it to be. This sets up your frame of reference. It doesn't change anything physically. Torque also depends on your choice of center.

1. What is the axis of rotation when a ball is rolling down a surface?

The axis of rotation is an imaginary line that passes through the center of the ball and remains stationary as the ball rolls down the surface. It is perpendicular to the direction of motion and is also referred to as the axis of symmetry.

2. How does the axis of rotation affect the motion of the rolling ball?

The axis of rotation determines the direction and speed of the ball's rotation. As the ball moves down the surface, the axis of rotation remains fixed, causing the ball to rotate around it. This rotation also helps to stabilize the ball's motion and keep it rolling in a straight line.

3. Can the axis of rotation change as the ball rolls down a surface?

No, the axis of rotation remains constant as long as the ball is rolling down the surface. It only changes if the ball experiences a force that shifts its axis, such as hitting an obstacle or changing direction.

4. Does the shape of the ball affect the location of the axis of rotation?

Yes, the shape of the ball can affect the location of the axis of rotation. For example, a perfectly spherical ball will have its axis of rotation at the center, while a non-spherical ball may have its axis of rotation slightly off-center.

5. How does friction impact the axis of rotation for a rolling ball?

Friction can cause the axis of rotation to shift slightly, especially if the surface is not completely smooth. This can result in the ball veering off its intended path and can also affect the speed and direction of rotation. However, the axis of rotation will still remain perpendicular to the direction of motion.