Answer:
1.97 m/s
Explanation:
The initial angular velocity of the blades is:
[tex]\omega_i = 70 rev/min = (70 rev/min) (\frac{2\pi rad/rev}{60 s/min})=7.33 rad/s[/tex]
The final angular velocity is
[tex]\omega_f = 0[/tex]
(since the blades come to a stop)
and the time taken is
t = 30 s
So, the angular acceleration of the blades is
[tex]\alpha = \frac{\omega_f-\omega_i}{t}=\frac{0-7.33 rad/s}{30 s}=-0.24 rad/s^2[/tex]
Now we can find the angular velocity of a blade after a time t=10 s:
[tex]\omega = \omega_i + \alpha t = 7.33 rad/s +(-0.24 rad/s^2)(10 s)=4.93 rad/s[/tex]
The distance of the tip of a blade from the centre is equal to half the diameter, so
[tex]r=\frac{80 cm}{2}=40 cm=0.40 m[/tex]
so, the speed of the tip of a blade is
[tex]v=\omega r=(4.93 rad/s)(0.4 m)=1.97 m/s[/tex]
The speed of the tip of a blade 10 s after the fan is turned off is 2.834m/s
The formula to be used to get the velocity of the tip of a blade is expressed as:
[tex]v=\omega r[/tex]
[tex]\omega[/tex] is the angular velocity
r is the radius of the ceiling fan
r = 80/2 = 40cm
To get the angular velocity, we will use the formula as shown:
[tex]\omega = \omega_0+\alpha t[/tex]
[tex]\alpha[/tex] is the angular acceleration
Get the angular acceleration:
[tex]\alpha = \frac{\omega_f-\omega_i}{t}\\\alpha = \frac{0-7.33}{30}\\\alpha =0.244rad/s^2[/tex]
Get the final angular velocity [tex]\omega[/tex]
[tex]\omega = 7.33 +(-0.244)(10)\\\omega =7.33-2.44\\\omega =7.086rad/s[/tex]
Get the required speed of the tip of the blade.
[tex]v = \omega r\\v=7.086 \times 0.4\\v=2.834m/s[/tex]
Hence the speed of the tip of a blade 10 s after the fan is turned off is 2.834m/s
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