What Is the Acceleration of the Bottom Point of a Rolling Object?

[jumbogala]

Homework Statement

The point on the bottom of a rolling object is instantaneously at rest (v = 0).

What is the acceleration of that point?

Homework Equations

The Attempt at a Solution

The formula given is a = \alphar, and r = 0. So I'm guessing it's zero as well, although that doesn't seem to make logical sense. Can anyone confirm that the answer is indeed zero?

(By the way, the equation for v = \omegar, and in this case r = 0. So r has to be zero for the equation for a as well.

 

[kuruman]

Confirmed instantaneously, zero. The entire wheel accelerates instantaneously relative to the point of contact. The point of contact is the only point on the wheel that is at rest with respect to an inertial frame, namely the floor.

 

[Doc Al]

Just because the point of contact is instantaneously at rest does not mean its acceleration is zero.

 

[jumbogala]

Seems like I'm getting conflicting answers...

If that's wrong, then how does the formula a = αr work in this case? I know in general that formula is useful for "converting" angular acceleration to linear acceleration, but how does it pertain to the bottom of a rolling object?

 

[Doc Al]

If that's wrong, then how does the formula a = αr work in this case? I know in general that formula is useful for "converting" angular acceleration to linear acceleration, but how does it pertain to the bottom of a rolling object?

I'd say that that formula won't help much here. It would tell you the tangential acceleration of some point with respect to the axis, but what's the angular acceleration in this case? Either way, it doesn't give the acceleration of the axis itself.

To find the acceleration of a point on the rim, consider the motion of that point with respect to the center of mass.

 

[jumbogala]

Why the center of mass? Isn't the axis in this case the bottom of the wheel?

 

[Doc Al]

Why the center of mass?

For one thing, we know its acceleration. Presumably the object is rolling at constant speed, so the acceleration of the center of mass is zero.

Isn't the axis in this case the bottom of the wheel?

Yes, the bottom of the wheel is the instantaneous axis of rotation. But we are trying to find the acceleration of that point, not of some other point about that axis.

And you didn't answer my question: What's the angular acceleration of this rolling object?

 

[kuruman]

Just because the point of contact is instantaneously at rest does not mean its acceleration is zero.

I agree, but in this case I think it does. The problem states

The point on the bottom of a rolling object is instantaneously at rest (v = 0).

It is implicit in that statement, that the reference frame, with respect to which the velocity is zero, is the inertial frame of the surface on which the object rolls. There is no other frame in which v = 0. The center of mass may do whatever it wants, but in the inertial frame of the surface (instantaneously), the acceleration of the point of contact is zero and its velocity is also zero.

 

[Doc Al]

It is implicit in that statement, that the reference frame, with respect to which the velocity is zero, is the inertial frame of the surface on which the object rolls. There is no other frame in which v = 0. The center of mass may do whatever it wants, but in the inertial frame of the surface (instantaneously), the acceleration of the point of contact is zero and its velocity is also zero.

(1) If the velocity and acceleration are both zero, how does it move?
(2) The acceleration is the same in any inertial frame, including the one in which the center of mass is at rest. Of course, from the center of mass frame it is easy to see that that point of contact is accelerating.

 

[vela]

But in the inertial frame of the surface (instantaneously), the acceleration of the point of contact is zero and its velocity is also zero.

I would say one component of the acceleration is zero, and I think this is true regardless of whether the center of mass is accelerating or not.

 

[Doc Al]

I would say one component of the acceleration is zero, and I think this is true regardless of whether the center of mass is accelerating or not.

Absolutely.

 

[kuruman]

(1) If the velocity and acceleration are both zero, how does it move?
(2) The acceleration is the same in any inertial frame, including the one in which the center of mass is at rest. Of course, from the center of mass frame it is easy to see that that point of contact is accelerating.

Of course. I set my priorities wrong when thinking about this.