Rolling Motion (Stone Gets Stuck in a Tire Going Forward)

[amandela]

Homework 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?

Relevant Equations

a = v^2/r

So I thought the stone would initially experience acceleration in the backward (leftward) direction then continually accelerate in the inward direction of the tire (i.e. upward then rightward then downward then leftward, etc.) as the tire moves forward. But the answer is immediately upward acceleration. Not sure why that is.

Thank you.

 

[.Scott]

You are probably thinking of its motion relative to the car.
The part of the tire that is in contact with the road is traveling backwards relative to the car.
All other parts of the tire have a forward component to their motion.
For example, the top of the tire is moving twice as fast as the car.

Any point on the surface of the tire will follow a cyclonic path:

 

[PeroK]

Homework 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?
Relevant Equations: a = v^2/r

So I thought the stone would initially experience acceleration in the backward (leftward) direction then continually accelerate in the inward direction of the tire (i.e. upward then rightward then downward then leftward, etc.) as the tire moves forward. But the answer is immediately upward acceleration. Not sure why that is.

Thank you.

You could always try calculating the acceleration.

 

[Orodruin]

The easiest way to consider this is to note that acceleration is invariant between inertial frames and then use the rest frame of the car. In that frame, the stone approaches at speed $v$ until it meets the tire and starts moving in a circle. At the bottom of the circular motion, the acceleration is clearly directed upwards.

 

[kuruman]

Any point on the surface of the tire will follow a cyclonic path:

Good heavens no. This is cyclonic

Your figure shows a cycloid.

 

[.Scott]

Your figure shows a cycloid.

I actually typed in cycloid. But then the spell-checker suggested cyclonic. I didn't look closely and assumed it wanted an adjective instead of a noun - something like "cycloidic" - so I agreed to the change.

I will now change it back ... and add "cycloid" to my computer dictionary.

 

[haruspex]

it wanted an adjective instead of a noun - something like "cycloidic"

The adjective is cycloidal.

 

[Orodruin]

I still perfer the circular motion in the car’s rest frame … also no linguistic discussion

 

[PeroK]

I still perfer the circular motion in the car’s rest frame … also no linguistic discussion

I was just thinking the same thing. Acceleration is invariant between inertial reference frames.

 

[phinds]

I actually typed in cycloid. But then the spell-checker suggested cyclonic. I didn't look closely and assumed it wanted an adjective instead of a noun - something like "cycloidic" - so I agreed to the change.

I will now change it back ... and add "cycloid" to my computer dictionary.

When typing on a computer, autocorrect is your worse enema.

 

[Orodruin]

Acceleration is invariant between inertial reference frames.

That’s what I said! Nobody seemed to notice

note that acceleration is invariant between inertial frames

When typing on a computer, autocorrect is your worse enema.

I’ll have you know enemas can be very liberating!