Physics Help Forum Calculate a derivation of a the angular speed of a solid body

 Feb 16th 2018, 06:34 AM #1 Junior Member   Join Date: Feb 2018 Posts: 2 Calculate a derivation of a the angular speed of a solid body I am practicing for a mechanics exam. Please find the problem and solution in the image: I am not able to reach the solution. The question is to derivate an expression of the angular speed to obtain the angular acceleration. The angular speed is a function of the rotation angle, so the problem, I think, it is a differential equation thing. So I tried with all derivation rules (implicit and so on), but I am unable to reach the solution. How do I derivate that? Thank you for the help Attached Thumbnails
 Feb 16th 2018, 08:28 AM #2 Senior Member   Join Date: Oct 2017 Location: Glasgow Posts: 303 Yes, you get a 1st order ODE. Some hints... We have $\displaystyle \dot{\phi} = \frac{2}{R} \sqrt{\frac{(M-\mu m_B gR)\phi}{(3m_A + 2m_B)}}$ If we let $\displaystyle A = \frac{2}{R} \sqrt{\frac{(M-\mu m_B gR)}{(3m_A + 2m_B)}}$ we can reduce the formula to $\displaystyle \dot{\phi} = A\phi^{1/2}$ This makes the ensuing algebra a bit more manageable. Then you can then perform another substitution of the form $\displaystyle u = \phi^{1/2}$ $\displaystyle \frac{du}{d \phi} = \frac{d\left(\phi^{1/2}\right)}{d\phi} = \frac{1}{2}\phi^{-1/2} = \frac{1}{2u}$ and form a chain rule to describe the ODE in terms of $\displaystyle u$ and $\displaystyle \dot{u}$. That ODE will then be much easier to solve. Have a go Give us a shout if you get stuck. Last edited by benit13; Feb 16th 2018 at 08:30 AM.
 Feb 18th 2018, 06:48 PM #3 Junior Member   Join Date: Feb 2018 Posts: 2 Thank you for your answer. I reviewed how to solve 1st order EDO. In this case, it seems that original EDO is separable, so I do not understand why it is necessary to do that substitution. However, I am confused because solving the EDO by the separable method, returns a complex expression. So I tried to follow your substitution but I am no able to separate terms and solve it. Could u please go on with your expressions to reach the solution? Thanks
Feb 19th 2018, 02:29 AM   #4
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Join Date: Oct 2017
Location: Glasgow
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 Originally Posted by luimucar Thank you for your answer. I reviewed how to solve 1st order EDO. In this case, it seems that original EDO is separable, so I do not understand why it is necessary to do that substitution. However, I am confused because solving the EDO by the separable method, returns a complex expression.
Ahh yes, the equation is separable, so my additional substitution is overkill. It's just a force of habit for me to substitute like that because I hate dealing with square roots in ODEs.

I don't know how you ended up with something complex though...

$\displaystyle \frac{d\phi}{dt} = A \phi^{1/2}$

$\displaystyle \int \phi^{-1/2} d \phi = \int A dt$

$\displaystyle 2 \phi^{1/2} = At + c$

$\displaystyle \phi = \frac{1}{2}(At+c)^2$

Then differentiate that twice to get $\displaystyle \ddot{\phi}$

 Tags angular, body, calculate, derivation, solid, speed

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