 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   Feb 16th 2018, 08:28 AM #2 Senior Member   Join Date: Oct 2017 Location: Glasgow Posts: 364 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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 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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post eli28 Thermodynamics and Fluid Mechanics 5 Nov 8th 2015 04:01 AM sadistprincess Periodic and Circular Motion 1 Oct 3rd 2013 06:02 AM man_in_motion Periodic and Circular Motion 2 Jul 27th 2010 12:44 PM Arrowstar Advanced Mechanics 0 Aug 8th 2009 05:42 PM climber123boy Advanced Mechanics 1 Mar 30th 2009 09:45 AM