# Angular motion

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 w. 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_pg(t) . Note the order of the subscripts: velocity of axle measured relative to the ground. Express your answer in terms of R, w,t and unit vectors x and/or y

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(t). Express the position vector of the pebble in terms of R,w ,t , and the unit vectors x and/or y of the xy coordinate system shown.

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

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

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The solution contains 3 pages including full derivations and graphs.