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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A wheel that is rotating at 33.3 rad/s is given an angular acceleration of 2.15 rad/s/s. Through what angle has the wheel turned when its angular speed reaches 72.0 rad/s?

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Q: "The blades in a blender rotate at a rate of 7290 rpm. When the motor is turned off during operation, the blades slow to rest in 3.67 s. What is the angular acceleration as the blades slow down? The initial rotation is in the positive direction."
The OTA's solution is attached. We both got an answer of 208 rad/s^2.

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