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Question Number 163053 by ajfour last updated on 03/Jan/22

Commented by ajfour last updated on 03/Jan/22

$Q . 162949 (r e v i s i t)$
	Answered by ajfour last updated on 04/Jan/22
	$e=((2R)/π)=((4a)/π) (2a=R) r^2 =(2acos φ)^2 +(3a+2asin φ)^2 dm_(sc) =2λadφ I_0 =((2aλ(6a)^2 )/3)+∫r^2 dm_(sc) =24λa^3 +2λa^3 ∫_0 ^( π) (13+12sin φ)dφ =24λa^3 +2λa^3 (13π+24) say m_(rod) =m=6λa ⇒ I_0 =ma^2 (12+((13π)/3))=(((36+13π)ma^2 )/3) M=(((π+3)m)/3) ⇒ I_0 =(((36+13π)MR^2 )/(4(π+3))) ✓ for center of mass: 2πaλ(((4a)/π))=(2πaλ+6aλ)s s=((4a)/(π+3)) p^2 =s^2 +9a^2 tan β=(s/(3a))=(4/(3(π+3))) Mgpcos (θ+β)=I_0 (((ωdω)/dθ)) ω^2 =2Mgp{sin (θ+β)−sin β} T=(√(I_0 /(2Mgp)))∫_0 ^( (π/2)−ϕ) (dθ/( (√(sin (θ+β)−sin β)))) where ϕ=sin^(−1) (2/3) T=(√((a(36+13π))/(4(3+π)g(√(((4/(π+3)))^2 +9)))))×∫★ ∫★=∫_0 ^( cos^(−1) (2/3)) (dθ/( (√(sin (θ+tan^(−1) (4/(3(π+3))))−(4/( (√(16+9(π+3)^2 )))))))) Sir, please check the answer this yields with yours..i will try too..$
	$e = \frac{2 R}{π} = \frac{4 a}{π} (2 a = R)$ $r^{2} = {(2 a \cos ϕ)}^{2} + {(3 a + 2 a \sin ϕ)}^{2}$ ${d m}_{s c} = 2 λ a d ϕ$ $I_{0} = \frac{2 a λ {(6 a)}^{2}}{3} + \int r^{2} {d m}_{s c}$ $= 24 λ a^{3} + 2 λ a^{3} \int_{0}^{π} (13 + 12 \sin ϕ) d ϕ$ $= 24 λ a^{3} + 2 λ a^{3} (13 π + 24)$ $s a y m_{r o d} = m = 6 λ a$ $\Rightarrow I_{0} = {m a}^{2} (12 + \frac{13 π}{3}) = \frac{(36 + 13 π) {m a}^{2}}{3}$ $M = \frac{(π + 3) m}{3}$ $\Rightarrow I_{0} = \frac{(36 + 13 π) {M R}^{2}}{4 (π + 3)}$ $✓$ $f o r c e n t e r o f m a s s :$ $2 π a λ (\frac{4 a}{π}) = (2 π a λ + 6 a λ) s$ $s = \frac{4 a}{π + 3}$ $p^{2} = s^{2} + 9 a^{2}$ $\tan β = \frac{s}{3 a} = \frac{4}{3 (π + 3)}$ $M g p \cos (θ + β) = I_{0} (\frac{ω d ω}{d θ})$ $ω^{2} = 2 M g p {\sin (θ + β) - \sin β}$ $T = \sqrt{\frac{I_{0}}{2 M g p}} \int_{0}^{\frac{π}{2} - φ} \frac{d θ}{\sqrt{\sin (θ + β) - \sin β}}$ $w h e r e φ = \sin^{- 1} \frac{2}{3}$ $T = \sqrt{\frac{a (36 + 13 π)}{4 (3 + π) g \sqrt{{(\frac{4}{π + 3})}^{2} + 9}}} \times \int ★$ $\int ★ = \int_{0}^{\cos^{- 1} \frac{2}{3}} \frac{d θ}{\sqrt{\sin (θ + \tan^{- 1} \frac{4}{3 (π + 3)}) - \frac{4}{\sqrt{16 + 9 {(π + 3)}^{2}}}}}$ $S i r, p l e a s e c h e c k t h e a n s w e r t h i s$ $y i e l d s w i t h y o u r s . . i w i l l t r y t o o . .$
	Commented by mr W last updated on 04/Jan/22

	$i g o t w i t h y o u r f o r m u l a T \approx 1.3626 \sqrt{\frac{R}{g}}$ $t h a t d i f f e r s f r o m m y r e s u l t .$
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