angular motion question

Question

a pulley driving a belt has a diameter of 360mm and is turning at 2700/pi revolutions per min

find the angular velocity of the pulley and the linear velocity of the belt assuming no slip occurs

can someone explain why is it 2700/pi first please

Answer 1

The ²⁷⁰⁰/_π is just how they chose to tell the RPM of the pulley, it makes it easier to do the calculations. They could've said the pulley rotates at 859.437 RPM but ²⁷⁰⁰/_π is better in this case.

v = ωr, so first we need to find ω

ω = 2πn where n is in ωrev/sec

²⁷⁰⁰/_π RPM / 60 sec = ²⁷⁰⁰/₆₀π rev/sec = ⁴⁵/_π rev/sec

ω = 2πn = 2π(⁴⁵/_π) = 90 rad/sec

Now we can find v

v = ωr = ω * ^d/₂

= 90 rad/sec * ³⁶⁰/₂ = 16,200 mm/sec = 16.2 ^m/_s