Question

Wheel A of radius $r_A=10\text{cm}$ is coupled by belt B to a wheel C of radius $r_c=25\text{cm}$. The angular speed of the wheel A is increased from rest at a constant rate of $1.6{\text{rad/s}}^2.$ Find the time needed for wheel C to reach an angular speed of $100\text{rev/min},$ assuming that the belt does not slip.

• A. 4 seconds
• B. 9 seconds
• C. 16 seconds
• D. 25 seconds

Answer 1

The correct option is C 16 seconds

Given that
Radius of wheel A, $r_A=10\text{cm}$
Radius of wheel C, $r_C=25\text{cm}$
Initial angular speed ${\omega }_0=0$
Angular acceleration of the wheel A $=1.6{\text{rad/s}}^2$
Let angular acceleration of wheel A be ${\alpha }_A$ and angular acceleration of wheel C be ${\alpha }_C$.
Since they are connected by the same belt,
Tangential acceleration of wheel A = tangential acceleration of wheel B
$\Rightarrow {\alpha }_A{r}_A={\alpha }_C{r}_C$
$\Rightarrow {\alpha }_C={\alpha }_A\left(\frac{r_A}{r_C}\right)=1.6\times \left(\frac{10}{25}\right)$
$\Rightarrow {\alpha }_C=0.64{\text{rad/s}}^2$
Now, ${\omega }_C={\alpha }_{C}t$
$\Rightarrow t=\frac{{\omega }_C}{{\alpha }_C}$
$\Rightarrow t=\frac{100\text{rev/min}}{0.64{\text{rad/sec}}^2}$
$\Rightarrow t=\frac{100\times \frac{2\pi }{60}\text{rad/sec}}{0.64{\text{rad/sec}}^2}$
$\Rightarrow t\approx 16\text{sec}$