Acceleration of stone caught in tire tread

[friendbobbiny]

1) Problem Statement:

A car travels forward with constant velocity. It goes over a small stone, which gets stuck in the groove of a tire. The initial acceleration of the stone, as it leaves the surface of the road, is
(A) vertically upward
(B) horizontally forward
(C) horizontally backward
(D) zero
(E) upward and forward, at approximately 45° to the horizontal

2) Relevant Formulas:

Objects in a circular path accelerate radially inwards

3) Attempt

a) The stone enters a circular path at the wheel's bottom. Thus its centripetal acceleration points upwards
b) The stone is accelerated from rest to the car's velocity. Thus, it experiences a translational acceleration rightwards.
c) The accelerations are both horizontal and upwards; therefore, e is the right answer

The answer key claims that the answer is a.

 

[paisiello2]

It says, "...as it leaves the surface of the road." What must be true in order for this to happen?

 

[friendbobbiny]

It says, "...as it leaves the surface of the road." What must be true in order for this to happen?

That leaving the surface depends only on the rock's upward movement? I'm not really sure about what must be true.

 

[paisiello2]

No, I am thinking if the rock is leaving the surface then it must already by stuck in the tire and therefore already moving with some velocity. So that only leaves one direction left for acceleration to happen. Agree?

 

[SammyS]

Presumably, the tire is not skidding. The part of the tire in contact with the road is momentarily at rest.

 

[haruspex]

No, I am thinking if the rock is leaving the surface then it must already by stuck in the tire and therefore already moving with some velocity. So that only leaves one direction left for acceleration to happen. Agree?

No. It starts at rest. Before having any velocity it must have a nonzero acceleration. This is what we are interested in finding.
Here's another way to think of it: when a wheel rolls, where is the instantaneous centre of rotation at a given instant?

 

[paisiello2]

Yes, you're right. I was thinking relative to the moving car.

 

[Brage]

If a car is going forwards, the bottom of its wheels will be going backwards (As every action has an equal and oposit reaction.. you can also easily visualize this by thinking of a rolling wheel). The trick is that although the bottom of the wheel is moving backwards (with a certain radial velocity), the whole wheel is moving forwards with the same velocity (if this was not true your wheel would be skidding). Think then about what is happening to the stone, i.e. what direction must it be traveling in as soon as it gets picked up? This is then the direction of acceleration of the stone.
If you need some help visualizing this wikipedia is always a good help ;)

 

[friendbobbiny]

So: Because the wheel picks up the stone at a point momentarily at rest, there is no rightwards acceleration?

Here's another way to think of it: when a wheel rolls, where is the instantaneous centre of rotation at a given instant?

The instantaneous center of rotation occurs at the wheel's center, no? What does this suggest?

 

[haruspex]

So: Because the wheel picks up the stone at a point momentarily at rest, there is no rightwards acceleration?

No, momentarily at rest does not mean there's no acceleration. A stone thrown straight up is momentarily at rest at its highest point, but the acceleration is constant.

The instantaneous center of rotation occurs at the wheel's center, no? What does this suggest?

No. The instantaneous centre of rotation is the point that is momentarily at rest. As you just observed, that is the part of the wheel in contact with the road. So at one moment the stone is the centre of rotation, but a fraction later it will be a point just beyond the stone. So, if the centre of rotation is on the ground, just a little way beyond the stone, and the stone is still almost at ground level, which way is the stone moving?

 

[SammyS]

From Wikipedia:

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage.

Here's the LINK to the Wikipedia article,​

Follow the above link for a very informative animation.

 

[friendbobbiny]

So, if the centre of rotation is on the ground, just a little way beyond the stone, and the stone is still almost at ground level, which way is the stone moving?

Wouldn't the stone be moving leftwards, even if slightly? My impression is that you wanted me to recognize that because "the stone is still almost at ground level", its positioning really hasn't changed. If it's positioning hasn't changed, its only acceleration is upwards?

 

[jbriggs444]

Wouldn't the stone be moving leftwards, even if slightly?

This gets into the mathematics of what is meant by "acceleration" which, in turn, gets into the definition of a "limit". As you focus in more and more tightly and consider shorter and shorter times and shorter and shorter distances that the stone has moved so far, the path of the stone over that fraction of a millisecond and fraction of a millimeter becomes more and more purely vertical.

Yes, you are correct that no matter how closely you focus in, there will be some horizontal movement (the curve of the cycloidal path will tend slightly forward). But the limit of the angle of the path (and of the acceleration along the path) as you look closer and closer at the instant when the stone leaves the road is vertical.

Another approach to this is to consider frames of reference...

Adopt a frame of reference in which the car is at rest. The centripetal acceleration of the stone at bottom-dead-center on the tire is purely vertical. Now translate to a frame in which the car is moving. You've added a constant velocity to the stone. Obviously this leaves its acceleration unchanged. So its acceleration remains vertical.