circular motion

Question

A rotating platform of radius R = 1.5 cm starts from rest and accelerates with uniform angular acceleration α = 22 rad/s².

a) write an expression that gives the centripetal acceleration on the edge of the platform as a function of time a_c(t)

b) write an expression for the time at which the magnitude of the centripetal and tangential accelerations at the edge will be equal

c) calculate the time t in seconds, for when the magnitudes of the centripetal and tangential accelerations are equal

Answer 1

Centripetal acceleration can be written as ω²r where ω is the angular velocity. This angular velocity is changing due to the angular acceleration and it can be written as ω = αt therefore, substituting this into the initial equation we get a_C(t) = (αt)²r = (22t)²(1.5) therefore a_C(t) = 726t² (in units of cm/s²)

If you want a generic equation then a_C(t) = α²t²r

Tangential acceleration (a_T) is rα so for the two to be equal means:

(αt)²r = rα

α²t²r = rα

αt²= 1

t² = 1/α

t = √(1/α) (you could rationalize this if you wanted)

Plugging in the value of α = 22 we get t = √(1/22) = 0.21 sec