The average angular speed of the grindstone is 10 rad/s
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Further explanation</h3>
<em>Let's recall </em><em>Angular Speed</em><em> formula as follows:</em>
![\boxed{ \omega = \omega_o + \alpha t }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5Comega%20%3D%20%5Comega_o%20%2B%20%5Calpha%20t%20%7D)
![\boxed{ \theta = \omega_o t + \frac{1}{2} \alpha t^2 }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5Ctheta%20%3D%20%5Comega_o%20t%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Calpha%20t%5E2%20%7D)
![\boxed{ \omega^2 = \omega_o^2 + 2 \alpha \theta }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5Comega%5E2%20%3D%20%5Comega_o%5E2%20%2B%202%20%5Calpha%20%5Ctheta%20%7D)
<em>where :</em>
<em>ω = final angular speed ( rad/s )</em>
<em>ω₀ = initial angular speed ( rad/s )</em>
<em>α = angular acceleration ( rad/s² )</em>
<em>t = elapsed time ( s )</em>
<em>θ = angular displacement ( rad )</em>
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<u>Given:</u>
radius of the grindstone = R = 4.0 m
initial angular speed = ω₀ = 8.0 rad/s
final angular speed = ω = 12 rad/s
elapsed time = t = 4.0 seconds
<u>Asked:</u>
average angular speed = ?
<u>Solution:</u>
<em>Firstly , we will calculate </em><em>angular displacement </em><em>as follows:</em>
![\theta = \frac{( \omega + \omega_o )}{2} t](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20%5Cfrac%7B%28%20%5Comega%20%2B%20%5Comega_o%20%29%7D%7B2%7D%20t%20)
![\theta = \frac{ ( 12 + 8.0 ) }{2} \times 4.0](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20%5Cfrac%7B%20%28%2012%20%2B%208.0%20%29%20%7D%7B2%7D%20%5Ctimes%204.0)
![\theta = 10 \times 4.0](https://tex.z-dn.net/?f=%5Ctheta%20%3D%2010%20%5Ctimes%204.0)
![\boxed {\theta = 40 \texttt{ rad}}](https://tex.z-dn.net/?f=%5Cboxed%20%7B%5Ctheta%20%3D%2040%20%5Ctexttt%7B%20rad%7D%7D)
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<em>Next , we could calculate the </em><em>average angular speed</em><em> as follows:</em>
![\texttt{average angular speed} = \theta \div t](https://tex.z-dn.net/?f=%5Ctexttt%7Baverage%20angular%20speed%7D%20%3D%20%5Ctheta%20%5Cdiv%20t)
![\texttt{average angular speed} = 40 \div 4.0](https://tex.z-dn.net/?f=%5Ctexttt%7Baverage%20angular%20speed%7D%20%3D%2040%20%5Cdiv%204.0)
![\boxed{\texttt{average angular speed} = 10 \texttt{ rad/s}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Ctexttt%7Baverage%20angular%20speed%7D%20%3D%2010%20%5Ctexttt%7B%20rad%2Fs%7D%7D)
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Learn more</h3>
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Answer details</h3>
Grade: High School
Subject: Physics
Chapter: Rotational Dynamics