Answer
The penny will start to slide off $2.93~seconds$ after the turntable is turned on.
Work Step by Step
At low angular speeds, the force of friction can provide the required centripetal force to keep the penny moving in a circle. We can find the angular speed when the force of static friction reaches its maximum possible value:
$F_f = m~a_r$
$mg~\mu_s = m~\omega^2~r$
$\omega = \sqrt{\frac{g~\mu_s}{r}}$
$\omega = \sqrt{\frac{(9.80~m/s^2)(0.350)}{0.100~m}}$
$\omega = 5.857~rad/s$
We can find the time $t$ when the turntable reaches this angular speed:
$\omega_f = \omega_0+\alpha~t$
$t = \frac{\omega_f - \omega_0}{\alpha}$
$t = \frac{5.857~rad/s - 0}{2.00~rad/s^2}$
$t = 2.93~s$
The penny will start to slide off $2.93~seconds$ after the turntable is turned on.
