A grinding wheel attained a velocity of 20 rad/sec in 5 sec starting from rest. Find the number of revolutions made by the wheel

The correct option is (25/π) rev/ sec (noting that the unit in the options is written as rev/sec, though the question asks for the total number of revolutions, which is unitless or simply revolutions).

To find the total number of revolutions made by the grinding wheel, we can follow these steps:

Step 1: Identify the given values
Initial angular velocity (${\omega }_0$) = $0\text{rad/sec}$ (since it starts from rest)
Final angular velocity ($\omega$) = $20\text{rad/sec}$
Time taken ($t$) = $5\text{sec}$

Step 2: Find the angular acceleration ($\alpha$)
Using the first equation of rotational motion:

$\omega ={\omega }_0+\alpha t$
Substituting the given values:

$20=0+\alpha \left(5\right)$
$\alpha =\frac{20}{5}=4{\text{rad/sec}}^2$

Step 3: Find the total angular displacement ($\theta$)
Using the second equation of rotational motion:

$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$
Substituting the values:

$\theta =\left(0\right)\left(5\right)+\frac{1}{2}\left(4\right){\left(5\right)}^2$
$\theta =2\times 25=50\text{radians}$

Step 4: Convert the angular displacement from radians to revolutions
Since one complete revolution is equal to $2\pi \text{radians}$, the number of revolutions ($N$) is given by:

$N=\frac{\theta }{2\pi }$
$N=\frac{50}{2\pi }=\frac{25}{\pi }\text{revolutions}$

Therefore, the total number of revolutions made by the wheel is $\frac{25}{\pi }$, which corresponds to the option written as (25/π) rev/ sec.