A grind stone starts from rest and has a constant angular acceleration of 4.0 rad/sec². The angular displacement and angular velocity, after 4 sec. will respectively be

The correct option is 32 rad, 16 rad/sec.

Step-by-step Explanation:

We are given the following parameters for the grindstone:
1. Initial angular velocity (${\omega }_0$) = 0 (since the grindstone starts from rest)
2. Constant angular acceleration ($\alpha$) = 4.0 rad/sec²
3. Time ($t$) = 4 sec

We need to find the angular displacement ($\theta$) and the final angular velocity ($\omega$) after 4 seconds.

1. Finding the Angular Displacement ($\theta$):
We can use the second equation of rotational motion:

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

Substituting the given values into the equation:

$\theta =\left(0\right)\left(4\right)+\frac{1}{2}\left(4.0\right){\left(4\right)}^2$

$\theta =0+2\left(16\right)$

$\theta =32\text{rad}$

Thus, the angular displacement is 32 rad.

2. Finding the Angular Velocity ($\omega$):
We can use the first equation of rotational motion:

$\omega ={\omega }_0+\alpha t$

Substituting the given values into the equation:

$\omega =0+\left(4.0\right)\left(4\right)$

$\omega =16\text{rad/sec}$

Thus, the angular velocity after 4 seconds is 16 rad/sec.

Therefore, the angular displacement and angular velocity after 4 seconds are 32 rad and 16 rad/sec, respectively.