1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Pebble in a Rolling Tire

  1. Oct 26, 2007 #1
    1. The problem statement, all variables and given/known data
    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.

    3. 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
     

    Attached Files:

    • 1.jpg
      1.jpg
      File size:
      12.2 KB
      Views:
      151
    • 2.jpg
      2.jpg
      File size:
      13.1 KB
      Views:
      146
  2. jcsd
  3. Oct 27, 2007 #2

    learningphysics

    User Avatar
    Homework Helper

    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).
     
  4. Oct 28, 2007 #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 ?
     
  5. Oct 28, 2007 #4
    bump .... anyone ?
     
  6. Oct 28, 2007 #5

    learningphysics

    User Avatar
    Homework Helper

    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: Oct 28, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?