Archived Rolling Motion, Pebble stuck in a Tire by Static Friction

[lila.haj]

Homework Statement

A 1.2 g pebble is stuck in a tread of a 0.76 m diameter automobile tire, held in place by static friction that can be at most 3.6 N. The car starts from rest and gradually accelerates on a straight road. How fast is the car moving when the pebble flies out of the tire tread? I know the answer is 17 m/s however I don't know how to get it.

Homework Equations

v=(2*\pi*R)/T
v=ωR
\alpha=τ/I
I=1/2*M*R²
F_c=(m*v²)/2

The Attempt at a Solution

I tried using the centripetal force equation and rearranged for velocity, using the maximum static friction as my force. I then divided that number by 2 to get the speed in the centre of the wheel (since the speed at the top of the wheel is twice the speed of the vehicle).

 

[CWatters]

I believe the approach suggested by the OP gives the correct answer.

F=(mv^2)/r
Rearrange for v...
V=SQRT(Fr/m)
= SQRT(3.6*0.38/0.0012)
= 33.76m/S

This is the Velocity of the top of the wheel. The Velocity of the car is half that..

33.76/2=16.88m/S

Book answer 17m/S.