The point on the bottom of a rolling object is instantaneously at rest. Acceleration?

In summary, the point on the bottom of a rolling object is instantaneously at rest with a velocity and acceleration of zero in the inertial frame of the surface on which it is rolling. The acceleration of the point of contact can be found by considering the motion of that point with respect to the center of mass. While the center of mass may have a non-zero acceleration, the point of contact will always have a component of acceleration that is zero.
  • #1
jumbogala
423
4

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 = [tex]\alpha[/tex]r, 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 = [tex]\omega[/tex]r, and in this case r = 0. So r has to be zero for the equation for a as well.
 
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  • #2


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.
 
  • #3


Just because the point of contact is instantaneously at rest does not mean its acceleration is zero.
 
  • #4


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?
 
  • #5


jumbogala said:
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.
 
  • #6


Why the center of mass? Isn't the axis in this case the bottom of the wheel?
 
  • #7


jumbogala said:
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? :wink:
 
  • #8


Doc Al said:
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.
 
  • #9


kuruman said:
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.
 
  • #10


kuruman said:
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.
 
  • #11


vela said:
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.
 
  • #12


Doc Al said:
(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.
 

1. What is the point on the bottom of a rolling object?

The point on the bottom of a rolling object refers to the part of the object that is in direct contact with the surface it is rolling on. This point is constantly changing as the object rolls, but at any given moment, it is the point that is closest to the ground.

2. What does it mean for the point to be instantaneously at rest?

When the point on the bottom of a rolling object is instantaneously at rest, it means that at that specific moment, it is not moving. This is because the point is in direct contact with the ground and is not moving relative to the surface it is rolling on.

3. Why does the point on the bottom of a rolling object have to be at rest?

The point on the bottom of a rolling object has to be at rest because of the principle of rolling motion. This principle states that for an object to roll, the point on the bottom must be at rest, otherwise the object would be sliding instead of rolling.

4. What is acceleration in relation to the point on the bottom of a rolling object?

Acceleration in relation to the point on the bottom of a rolling object refers to the rate of change of the object's velocity at that specific point. This means that the point is either speeding up or slowing down, depending on the direction of the acceleration.

5. How does the point on the bottom of a rolling object affect its overall acceleration?

The point on the bottom of a rolling object plays a crucial role in its overall acceleration. As the point is at rest, it allows the object to rotate smoothly and maintain its rolling motion. Any changes in the acceleration of the point will affect the object's overall acceleration and motion.

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