In figure. wheel $A$ of radius $r_A=10cm$ is coupled by belt $B$ to wheel $C$ of radius $r_C=25cm$. The angular speed of wheel $A$ is increased from rest at a constant rate, of $1.6rad/s^2.$ Find the time needed for wheel $C$ to reach an angular speed of $12.8rad/s,$ assuming the belt does not slip.

The correct option is C $20s$
Step 1:Find the angular acceleration of the wheel $C.$
Formula used: $a_t=ra$
Given, radius of wheel $A,r_A=10cm$
Radius of wheel $C,r_C=25cm$
Angular acceleration of wheel $A,$
$a_A=1.6rad/s^2$
if the belt does not slip, the linear speeds at the two rims must be equal, since the belt does not slip, a point on the rim of wheel $C$ has the same tangential acceleration as a point on the rim of wheel $A.$
$a_A{r}_A=a_c{r}_c$
$a_c=\left(\frac{r_A}{r_C}{a}_A\right)$
$a_c=\left(\frac{10}{25}\right)\left(1.6\right)$
$0.64rad/s^2$

Step 2:Find the time taken by wheel $C$ to reach an angular speed of $12.8rad/s.$
Formula used:${\omega }_c=12.8rad/s$
Time needed to achieve this angular speed,

$t=\frac{{\omega }_c}{a_c}$

$t=\frac{12.8}{0.64}=20s$
Final answer: $\left(c\right)$