The vertical speed of the rod varies inversely as the rotational speed for a given number of complete rotations
The equation that gives the vertical velocity, v₀ is ![\, \underline{v_0= \dfrac{ g \cdot \pi \cdot n}{ \omega_0 }}](https://tex.z-dn.net/?f=%5C%2C%20%5Cunderline%7Bv_0%3D%20%20%5Cdfrac%7B%20g%20%5Ccdot%20%5Cpi%20%5Ccdot%20n%7D%7B%20%20%5Comega_0%20%20%20%7D%7D)
Reason:
<em>From a similar question, the given parameters appear to be correctly given as follows;</em>
<em>The vertical speed of the rod = v₀</em>
<em>Angular velocity of the = ω₀</em>
Required;
The value of, v₀, so that the rod has made exactly, <em>n</em>, number of turns when it returns to his hand
Where;
n = An integer
Solution;
The time the rod spends in the air is given as follows;
![Height, \ h = u \cdot t - \dfrac{1}{2} \cdot g \cdot t^2](https://tex.z-dn.net/?f=Height%2C%20%5C%20h%20%3D%20u%20%5Ccdot%20t%20-%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20g%20%5Ccdot%20t%5E2)
Where;
g = Acceleration due to gravity
t = Time of motion
When the rod returns to his hand, we have, h = 0, therefore;
![0 = u \cdot t - \dfrac{1}{2} \cdot g \cdot t^2](https://tex.z-dn.net/?f=0%20%3D%20u%20%5Ccdot%20t%20-%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20g%20%5Ccdot%20t%5E2)
![u \cdot t = \dfrac{1}{2} \cdot g \cdot t^2](https://tex.z-dn.net/?f=u%20%5Ccdot%20t%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20g%20%5Ccdot%20t%5E2)
![t^2 = \dfrac{u \cdot t }{\dfrac{1}{2} \cdot g } = \dfrac{2 \cdot u \cdot t }{g }](https://tex.z-dn.net/?f=t%5E2%20%3D%20%20%5Cdfrac%7Bu%20%5Ccdot%20t%20%7D%7B%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20g%20%7D%20%3D%20%20%20%5Cdfrac%7B2%20%5Ccdot%20u%20%5Ccdot%20t%20%7D%7Bg%20%7D)
![Time, \ t = \dfrac{2 \cdot u }{g }](https://tex.z-dn.net/?f=Time%2C%20%5C%20t%20%3D%20%20%20%20%5Cdfrac%7B2%20%5Ccdot%20u%20%20%7D%7Bg%20%7D)
We have;
![Angular \ velocity, \ \omega_0 = \dfrac{ 2 \cdot \pi \cdot n}{t} \ (required)](https://tex.z-dn.net/?f=Angular%20%5C%20velocity%2C%20%5C%20%5Comega_0%20%3D%20%5Cdfrac%7B%202%20%5Ccdot%20%5Cpi%20%5Ccdot%20n%7D%7Bt%7D%20%5C%20%28required%29)
![\omega_0 = \dfrac{ 2 \cdot \pi \cdot n}{ \dfrac{2 \cdot v_0 }{g }} = \dfrac{ g \cdot \pi \cdot n}{ v_0 }](https://tex.z-dn.net/?f=%5Comega_0%20%3D%20%5Cdfrac%7B%202%20%5Ccdot%20%5Cpi%20%5Ccdot%20n%7D%7B%20%20%5Cdfrac%7B2%20%5Ccdot%20v_0%20%20%7D%7Bg%20%7D%7D%20%3D%20%20%5Cdfrac%7B%20g%20%5Ccdot%20%5Cpi%20%5Ccdot%20n%7D%7B%20%20v_0%20%7D)
Therefore;
![The \ vertical \ velocity, \, \underline{v_0= \dfrac{ g \cdot \pi \cdot n}{ \omega_0 }}](https://tex.z-dn.net/?f=The%20%5C%20vertical%20%5C%20velocity%2C%20%5C%2C%20%5Cunderline%7Bv_0%3D%20%20%5Cdfrac%7B%20g%20%5Ccdot%20%5Cpi%20%5Ccdot%20n%7D%7B%20%20%5Comega_0%20%20%20%7D%7D)
Learn more here:
brainly.com/question/14140053