a) The length of the arm of the centrifuge is 10.9 m
b) The angular acceleration is ![2.7 rad/s^2](https://tex.z-dn.net/?f=2.7%20rad%2Fs%5E2)
Explanation:
a)
In a uniform circular motion, the centripetal acceleration is given by
![a_c=\omega^2 r](https://tex.z-dn.net/?f=a_c%3D%5Comega%5E2%20r)
where:
is the angular speed of the circular motion
r is the radius of the circle
For the centrifuge in this problem, we have:
is the angular speed
The centripetal acceleration is 3.2 times the acceleration due to gravity (
), so:
![a_c=3.2 g = 3.2(9.8)=31.4 m/s^2](https://tex.z-dn.net/?f=a_c%3D3.2%20g%20%3D%203.2%289.8%29%3D31.4%20m%2Fs%5E2)
Therefore, we can re-arrange the previous equation to find r, the radius of the circle (which corresponds to the length of the arm of the centrifuge):
![r=\frac{a_c}{\omega^2}=\frac{31.4}{1.7^2}=10.9 m](https://tex.z-dn.net/?f=r%3D%5Cfrac%7Ba_c%7D%7B%5Comega%5E2%7D%3D%5Cfrac%7B31.4%7D%7B1.7%5E2%7D%3D10.9%20m)
b)
In the second part of the exercise, the centrifuge speeds up from an initial angular speed of 0 to a final angular speed of 1.7 rad/s. The total acceleration experienced at the final moment is
![a=4.4 g](https://tex.z-dn.net/?f=a%3D4.4%20g)
So, 4.4 times the acceleration due to gravity.
The total acceleration is the resultant of the centripetal acceleration (
) and the tangential acceleration (
):
![a=\sqrt{a_c^2+a_t^2}](https://tex.z-dn.net/?f=a%3D%5Csqrt%7Ba_c%5E2%2Ba_t%5E2%7D)
We know that:
a = 4.4g
![a_c = 3.2 g](https://tex.z-dn.net/?f=a_c%20%3D%203.2%20g)
So, we can find the tangential acceleration:
![a_t = \sqrt{a^2-a_c^2}=\sqrt{(4.4g)^2-(3.2g)^2}=29.6 m/s^2](https://tex.z-dn.net/?f=a_t%20%3D%20%5Csqrt%7Ba%5E2-a_c%5E2%7D%3D%5Csqrt%7B%284.4g%29%5E2-%283.2g%29%5E2%7D%3D29.6%20m%2Fs%5E2)
The angular acceleration is related to the tangential acceleration by
![\alpha = \frac{a_t}{r}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7Ba_t%7D%7Br%7D)
where r = 10.9 m is the length of the centrifuge. Substituting,
![\alpha = \frac{29.6}{10.9}=2.7 rad/s^2](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B29.6%7D%7B10.9%7D%3D2.7%20rad%2Fs%5E2)
Learn more about centripetal and angular acceleration here:
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