What initial speed must the top be given in order to spin for at least 1 minute?

the angular acceleration of an object can be described via

angular acceleration=(change in angular velocity)/time

ang accel = (final ang vel - initial ang vel)/time

the ang accel = -1.3 rad/s/s (note the minus sign)

let's convert 3.5 rev/s to rad /s

3.5 rev/s = 3.5 rev/s x 2pi rad/ rev x 1 min/60s = 0.37rad/s

so, for the object to be in motion for a minute, losing speed at the rate of 1.3 rad/s/s and remaining above 0.37rad/s, the initial speed is

initial ang vel = final ang vel -(-1.3rad/s)x 60 s

initial ang vel = 0.37rad/s + 78rad/s = 78.37 rad/s

this is the same as 748.8rev/min

final angular velocity wF = 3.5 x 2 pi = 21.991 rad/ sec

time = t = 60 sec

angular acceleration = - 1.3 rad / sec^2

Initial angular velocity = vI

final angular velocity = Initial angular velocity + angular acceleration x time

21.991 = vI - 1.3 x 60

vI = 99.991 rad /sec ans