Question

"a grindstone of radius 4.0 m is initially spinning with an angular speed of 8.0 rad/s. the angular speed is then increased to 12 rad/s over the next 4.0 seconds. assume that the angular acceleration is constant. what is the average angular speed of the grindstone?"

Answer 1

The average angular speed of the grindstone is 10 rad/s

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Further explanation

Let's recall Angular Speed formula as follows:

$\boxed{ \omega = \omega_o + \alpha t }$

$\boxed{ \theta = \omega_o t + \frac{1}{2} \alpha t^2 }$

$\boxed{ \omega^2 = \omega_o^2 + 2 \alpha \theta }$

$\boxed{ \theta = \frac{( \omega + \omega_o )}{2} t }$

where :

ω = final angular speed ( rad/s )

ω₀ = initial angular speed ( rad/s )

α = angular acceleration ( rad/s² )

t = elapsed time ( s )

θ = angular displacement ( rad )

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Given:

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

Asked:

average angular speed = ?

Solution:

Firstly , we will calculate angular displacement as follows:

$\theta = \frac{( \omega + \omega_o )}{2} t$

$\theta = \frac{ ( 12 + 8.0 ) }{2} \times 4.0$

$\theta = 10 \times 4.0$

$\boxed {\theta = 40 \texttt{ rad}}$

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Next , we could calculate the average angular speed as follows:

$\texttt{average angular speed} = \theta \div t$

$\texttt{average angular speed} = 40 \div 4.0$

$\boxed{\texttt{average angular speed} = 10 \texttt{ rad/s}}$

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Answer details

Grade: High School

Subject: Physics

Chapter: Rotational Dynamics

Answer 2

We know the initial and final angular speed of the grindston, and also the time t=4.0 s, so by using these data we can find the angular acceleration of the grindstone:
$\alpha= \frac{\omega_f - \omega_i}{t}= \frac{12.0 rad/s-8.0 rad/s}{4.0 s}=1 rad/s^2$

And then we can find the total angle covered by the grindstone during the time t=4.0 s:
$\theta= \omega_i t + \frac{1}{2} \alpha t^2 = (8.0 rad/s)(4.0 s)+ \frac{1}{2}(1.0 rad/s^2)(4.0 s)^2=40 rad$

The average angular speed of the grindstone is the angular speed it would have when covering the angular distance of 40 rad in 4.0 s in uniform angular motion, so
$\omega_{avg} = \frac{\theta}{t}= \frac{40 rad}{4.0 s}=10 rad/s$