Your roommate is working on his bicycle and has the bike upside down. he spins the 60 cm
diameter wheel, and a pebble in the tread goes by 3 times every second.
what is the pebbles speed and

Diameter = 60 cm, Radius = 60/2 = 30 cm = 30/100 = 0.3 m.

The pebble in the tread goes by 3 times every second.

This is the same as 3 times per second.

Recall the unit of frequency is Hertz or per second, s⁻¹

So 3 times per second, Frequency, f = 3s⁻¹ or 3 Hertz

For angular motion:

Angular speed, ω = 2πf

                         = 2*π*3

                         = 6π   rad/s

Linear speed, v = ωr =  6π * 0.3 = 1.8π m/s

Linear acceleration, a = v² / r

                               a = 1.8π * 1.8π / 0.3 = 10.8π²   m/s²

Angular acceleration α = a/r  = 10.8π² / 0.3 = 36π² rad/s²

Angular speed = 6π rad/s ≈ 18.840 rad/s

The linear speed of the pebble = 1.8π  m/s ≈ 5.655 m/s

The angular acceleration = 36π² rad/s² ≈ 355.306 rad/s²

The linear acceleration of the pebble = 10.8π²  m/s ≈ 106.592 m/s²