The fan's angular acceleration assumed constant is -0.42rad/s²
The time it takes the fan to come to a complete stop is 211.90secs
In order to get the fan's angular acceleration, we will use the equation of rotation is expressed as:
ω² = ω₀² + 2 α θ where;
ω is the final angular velocity
ω₀ is the initial angular velocity
α is the angular acceleration
θ is the angle
Since the body comes to rest, hence ω = 0
θ = 1500 x 2 x π
θ = 9424.78 rad
Substitute the given parameters to get the angular acceleration
[tex]0 = 89^2 + 2 \times \alpha \times 9424.78\\-7921=18,849.56\alpha\\\alpha=\frac{-7921}{18849.56}\\\alpha= -0.42rad/s^2[/tex]
b) Next is to get the time it takes for the fan to stop
[tex]\omega = \omega_0+\alpha t\\0=89-0.42t\\0.42t=89\\t=\frac{89}{0.42}\\t= 211.90secs[/tex]
Hence the time it takes the fan to come to a complete stop is 211.90secs
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The angular acceleration of fan is [tex]-16.58 \;\rm rad/s^{2}[/tex].
The fan will take 5.36 seconds to stop completely.
Given data:
The number of rotation made by fan per minute is, [tex]N = 850 \;\rm rpm[/tex].
The number of revolutions is, [tex]n = 1500[/tex].
(a)
The angular displacement made by the fan is,
[tex]\theta = \dfrac{n}{2 \pi} \\\theta = \dfrac{1500}{2 \pi}\\\theta = 238.73 \;\rm rad[/tex]
And, angular speed at initial is,
[tex]\omega_{i}=\dfrac{2\pi N}{60} \\\omega_{i}=\dfrac{2\pi \times 850}{60}\\\omega_{i}=89 \;\rm rad/s[/tex]
Now, apply the third rotational equation of motion as,
[tex]\omega^{2}_{f}=\omega^{2}_{i}+2 \alpha \theta\\0^{2}=89^{2}+2 \times \alpha \times 238.73\\\alpha =-16.58 \;\rm rad/s^{2}[/tex]
Thus, the angular acceleration of fan is [tex]-16.58 \;\rm rad/s^{2}[/tex].
(b)
Applying the first rotational equation of motion to obtain the time taken to stop as,
[tex]\omega_{f}=\omega_{i}+\alpha t\\0= 89+(-16.58) t\\t = 5.36 \;\rm s[/tex]
Thus, the fan will take 5.36 seconds to stop completely.
Learn more about the rotational equation of motion here:
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