Pebble in a Rolling Tire

In summary: So, yeah... just take the derivative of this. In summary, the problem involves finding the coordinates of a pebble stuck in a rolling tire with angular velocity omega. The tire is rolling without slipping on the ground and has an outer radius of R. The velocity of the axle of the tire can be found using v_vec_ag(t) = -Rw\hat{x}. The position vector of the pebble relative to the initial point of contact between the wheel and ground at a time t can be expressed as \vec{r} = (-Rsin(wt)-Rwt)\hat{x} + (Rcos(wt)+R)\hat{y}, and the velocity vector of the pebble with respect to a fixed point
  • #1
HeLLz aNgeL
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Homework Statement


You are to find the coordinates of a pebble stuck in the tread of a rolling tire that is rotating counterclockwise (i.e., in the positive sense) with angular velocity omega. The tire rolls without slipping on the ground (which is at y = 0 ). The outer radius of the tire is R. At time t = 0 , the pebble is at the top of the tire, as shown.

a). Find the velocity of the axle of the tire relative to a fixed point on the ground, v_vec_ag(t). Note the order of the subscripts: velocity of axle measured relative to the ground. Express your answer in terms of R, omega, and x_unit and/or y_unit.

The pebble and tire have now rolled as shown in the figure. View Figure Answer the following questions for t>0.

b). Find the position vector of the pebble relative to the initial point of contact between the wheel and ground at a time t, r_pg_vec(t).
Express the position vector of the pebble in terms of R, omega, t, and the unit vectors x_unit and/or y_unit of the xy coordinate system shown.


c). Find v_vec_pg(t), the velocity vector of the pebble with respect to a fixed point on the ground, in terms of the unit vectors x_unit and y_unit of the xy coordinate system shown.
Express the velocity vector in terms of R, omega, t, and x_unit and/or y_unit.


d) Now find a_vec_pg(t), the acceleration vector of the pebble with respect to a fixed point on the ground.
Express your answer in terms of R, omega, t and x_unit and/or y_unit of the xy coordinate system shown.

The Attempt at a Solution



ok, for the first one, i know v = rw, but how do i write it in terms of vectors, v =rwx ?

and for part b, i have no clue how i should be approaching this ... just seems too weird :S
 

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  • #2
I think for part a), what they want is:

v_vec_ag(t) = -Rw[tex]\hat{x}[/tex]... ie -Rw times a unit vector in the x-direction.

For part b), suppose the wheel was just rotating without any translation... (ie it is just rotating in place). What is the position vector relative to the initial point of contact? use sin, cos etc... the angle through which the wheel has rotated is wt...

then how does this position vector change when you take into account the horizontal velocity of the tire?

c) take the derivative of b).

d) take the derivative of c).
 
  • #3
ok, so i know its Rsinwt(x) + RCoswt(y) and the initial position was just R(y)

so, relative to the initial position it should be Rsinwt(x)+(R(y)-RCoswt(y))

doesnt sound right, does it ?
 
  • #4
bump ... anyone ?
 
  • #5
HeLLz aNgeL said:
ok, so i know its Rsinwt(x) + RCoswt(y) and the initial position was just R(y)

so, relative to the initial position it should be Rsinwt(x)+(R(y)-RCoswt(y))

doesnt sound right, does it ?

Well... if translation wasn't there... here's what I get:

[tex]\vec{r} = -Rsin(wt)\hat{x} + (Rcos(wt)+R)\hat{y}[/tex]

Now add the translation... [tex]-Rwt\hat{x}[/tex]

so it comes out to:

[tex]\vec{r} = (-Rsin(wt)-Rwt)\hat{x} + (Rcos(wt)+R)\hat{y}[/tex]
 
Last edited:

1. What is the "pebble in a rolling tire" phenomenon?

The "pebble in a rolling tire" phenomenon refers to the behavior of a pebble or small object that gets stuck in the tread of a rolling tire and causes the tire to wobble or vibrate.

2. Why does a pebble in a rolling tire cause the tire to wobble?

The wobbling effect is due to the uneven distribution of weight caused by the pebble being stuck in the tread of the tire. This disrupts the balance of the tire and causes it to wobble as it rolls.

3. Can a pebble in a rolling tire cause damage to the tire?

Yes, if a pebble is stuck in the tread of a tire for a prolonged period of time, it can cause damage to the tire. The constant wobbling can wear down the tread and potentially lead to a flat tire or other issues.

4. How can I prevent a pebble from getting stuck in my tire?

Regularly checking your tires for any debris or objects stuck in the tread can help prevent a pebble from getting stuck. Avoiding driving on rough or rocky terrain can also reduce the chances of picking up a pebble.

5. Is the "pebble in a rolling tire" phenomenon dangerous?

In most cases, the wobbling caused by a pebble in a rolling tire is not dangerous and will not affect the overall performance of the vehicle. However, it is important to address the issue and remove the pebble to prevent potential damage to the tire.

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