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integration – Convergence of integral and summation for Time taken for full revolution round upright coterie Answer

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integration – Convergence of integral and summation for Time taken for full revolution round upright coterie

I wish to discover Time taken to finish Vertical round movement by Particle of mass $m$

So I proceed as follows

Applying labor power theorem

$$-mgR(1-cos(theta)=frac12 mv^2-frac12 mu^2$$

$$implies v^2=u^2-2gR(1-costheta)$$

$$implies omega=frac{sqrt{u^2-2gR(1-costheta)}}{R}$$

$$implies frac {dtheta}{dt}=frac{sqrt{u^2-2gR(1-costheta)}}{R}$$

$$implies frac {dtheta}{sqrt{u^2-2gR(1-costheta)}}=frac{dt}{R}$$

Let T breathe the time taken for full revolution

Now, Integrating either side

$$int_{0}^{2pi}frac {dtheta}{sqrt{u^2-2gR(1-costheta)}}=int_{0}^{T}frac{dt}{R}$$

$$implies T=2Rint_{0}^{pi}frac{dtheta}{sqrt{u^2-4gRsin^2theta}}$$

$$implies T=frac{2R}{u}int_{0}^{pi}frac{dtheta}{sqrt{1-frac{4gR}{u^2}sin^2theta}}$$

On evaluating this elliptic integral we get,

$$T=sum_{n=0}^{infty} left(frac{2}{u}privilege)^{2n+1}left(frac{(2n-1)!!}{2^n n!}privilege)^2 (gR)^n$$

Here $T$ denotes Time taken for full revolution. Block will full complete coterie provided that $ugeq sqrt{5gR}$. So my query is $T$ will breathe Real solely when $ugeq sqrt{5gR}$ so This situation ought to live in integral in addition to summation for $T$ however I’m not capable of behold it. Also Wolfram alpha evaluates integral provided that $ugeq sqrt{5gR}$

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