Rotation and Air Resistance

Since no specific shape is given, I will analyze a small square located a distance \( r \) from the axis (shown in red in the diagram).

Moment of inertia:

I=Mr2 I = M r^2

Pressure, force, and torque (let ω \omega be the angular frequency):

P=kv=krωF=PS=kSrωτ=Fr=kSr2ω P = k v = k r \omega \\ F = P S = k S r \omega \\ \tau = F r = k S r^2 \omega

Rotational Newton's 2nd Law:

τ=Iω˙kSr2ω=Mr2ω˙ω˙=kSMω -\tau = I \dot{\omega} \\ -k S r^2 \omega = M r^2 \dot{\omega} \\ \dot{\omega} = - \frac{k S }{M} \omega

This corresponds to exponential decay of the angular frequency:

ω=ω0ekSt/M \omega = \omega_0 \, e^{-k S t/M}

The angle swept out over infinite time is:

θ=0ω0ekSt/Mdt=ω0MkS \theta = \int_0^{\infty} \omega_0 \, e^{-k S t/M} \, dt = \frac{\omega_0 M}{k S}

And the number of revolutions is therefore:

Nrev=θ2π=ω0M2πkS N_{rev} = \frac{\theta}{2 \pi} = \frac{\omega_0 M}{2 \pi k S}

Note by Steven Chase
1 month ago

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@Lil Doug Here it is

Steven Chase - 1 month ago

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@Steven Chase Thank you so much.

Talulah Riley - 1 month ago

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@Steven Chase but one thing I want to ask how can we still call if no specific shape is given?

Talulah Riley - 1 month ago

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I assume that means that the result holds for any shape. We could probably analyze a spinning disk and come up with the same result.

Steven Chase - 1 month ago

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@Steven Chase Very nice thanks .
By the way,Today I just read one fact on internet , and the fact is “If the people whose name starts with S helps the people whose name starts with N , then the people with S names get a lot of benefit in the Physics subject.

Talulah Riley - 1 month ago

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@Steven Chase Take a look on this problem whenever you will be free, thanks in advance.
If possible post a note. If it takes time to write latex , please use pen and page.
hope I am not disturbing you.

Talulah Riley - 1 month ago

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