Answer
$a = 4.02~m/s^2$
Work Step by Step
Since the sphere is rolling without slipping, the force of static friction directed up the incline provides the torque to rotate the sphere. We can find an expression for the force of friction $F_f$:
$\tau = I~\alpha$
$R~F_f = I~\alpha$
$R~F_f = I~\frac{a}{R}$
$F_f = I~\frac{a}{R^2}$
$F_f = (\frac{2}{5}MR^2)(\frac{a}{R^2})$
$F_f = \frac{2}{5}Ma$
We can find the acceleration of the sphere:
$Mg~sin~\theta-F_f = Ma$
$Mg~sin~\theta = Ma+\frac{2}{5}Ma$
$Mg~sin~\theta = \frac{7}{5}Ma$
$a = \frac{5}{7}~g~sin~\theta$
$a = \frac{5}{7}~(9.80~m/s^2)~sin~35^{\circ}$
$a = 4.02~m/s^2$
