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Sagot :
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 [tex]\, \underline{v_0= \dfrac{ g \cdot \pi \cdot n}{ \omega_0 }}[/tex]
Reason:
From a similar question, the given parameters appear to be correctly given as follows;
The vertical speed of the rod = v₀
Angular velocity of the = ω₀
Required;
The value of, v₀, so that the rod has made exactly, n, 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;
[tex]Height, \ h = u \cdot t - \dfrac{1}{2} \cdot g \cdot t^2[/tex]
Where;
g = Acceleration due to gravity
t = Time of motion
When the rod returns to his hand, we have, h = 0, therefore;
[tex]0 = u \cdot t - \dfrac{1}{2} \cdot g \cdot t^2[/tex]
[tex]u \cdot t = \dfrac{1}{2} \cdot g \cdot t^2[/tex]
[tex]t^2 = \dfrac{u \cdot t }{\dfrac{1}{2} \cdot g } = \dfrac{2 \cdot u \cdot t }{g }[/tex]
[tex]Time, \ t = \dfrac{2 \cdot u }{g }[/tex]
We have;
[tex]Angular \ velocity, \ \omega_0 = \dfrac{ 2 \cdot \pi \cdot n}{t} \ (required)[/tex]
[tex]\omega_0 = \dfrac{ 2 \cdot \pi \cdot n}{ \dfrac{2 \cdot v_0 }{g }} = \dfrac{ g \cdot \pi \cdot n}{ v_0 }[/tex]
Therefore;
[tex]The \ vertical \ velocity, \, \underline{v_0= \dfrac{ g \cdot \pi \cdot n}{ \omega_0 }}[/tex]
Learn more here:
https://brainly.com/question/14140053
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