Problem: A disk initially rotating at 4 radians/second rotates through an angle of 14 radians while accelerating at 1 radians/second ^ 2. How long does this take, and what is the final angular velocity?
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Solution: There is no simple way to reason this problem out. So we use the analog to the equation of uniformly accelerating linear motion which relates vf, v0, a and ds:
corresponding equation for linear motion: vf ^ 2 = v0 ^ 2 + 2a (ds).
The angular equation would be expressed as
equation in terms of angular quantites: `omegaf^2 =`omega0^2 + 2 `alpha `dTheta.
We obtain final angular velocity
`omegaf = `sqrt(`omega0^2 + 2 `alpha `dTheta) = +- `sqrt [( 4 rad/sec) ^ 2+ 2( 1 rad/sec ^ 2)( 14 rad)] = +- 6.633 rad/sec.
We can reject the negative result, which is meaningless in this context, and we conclude that the final angular velocity is 6.633 rad/sec.
From the initial and final angular velocities, we obtain
average anglular velocity = 5.316,
from which we can calculate the time required to rotate through 14 radians. The result is
`dt = ( 14 rad)/( 5.316 rad/sec) = 2.633 sec.