Question and Answers Forum

All Questions      Topic List

Oscillation and Waves Questions

Previous in All Question      Next in All Question      

Previous in Oscillation and Waves      Next in Oscillation and Waves      

Question Number 175059 by mr W last updated on 17/Aug/22

Commented by mr W last updated on 17/Aug/22

$$asolidhemispherewithmassMand$$$$radiusRisconnectedwithathinrod$$$$withmassmandlengthLatthe$$$$center.$$$$findtheperiodofsmalloscillations.$$

Commented by mr W last updated on 18/Aug/22

Answered by mr W last updated on 18/Aug/22

Commented by mr W last updated on 18/Aug/22

$$OA=\frac{L}{2}$$$$OB=\frac{3R}{8}$$$$e=OC$$$$e+\frac{L}{2}=\frac{M}{m+M}\times (\frac{L}{2}+\frac{3R}{8})$$$$e=\frac{M}{m+M}\times (\frac{L}{2}+\frac{3R}{8})-\frac{L}{2}>0$$$$I_P=\frac{{mL}^2}{12}+m{(R+\frac{L}{2})}^2+\frac{83{MR}^2}{320}+M{\left(\frac{5R}{8}\right)}^2$$$$I_P=m(\frac{L^2}{3}+R^2+RL)+\frac{13{MR}^2}{20}$$

Commented by mr W last updated on 18/Aug/22

Commented by mr W last updated on 18/Aug/22

$$I_p{\left(\frac{d\theta }{dt}\right)}^2+(m+M)g(R-e\cos \theta )=E=constant$$$$2I_p\left(\frac{d\theta }{dt}\right)\frac{d^2\theta }{{dt}^2}+(m+M)ge\sin \theta \frac{d\theta }{dt}=0$$$$2I_p\frac{d^2\theta }{{dt}^2}+(m+M)ge\sin \theta =0$$$$forsmall\theta ,\sin \theta \approx \theta$$$$\Rightarrow \frac{d^2\theta }{{dt}^2}+\frac{(m+M)eg}{2I_P}\theta =0\to d.e.ofs.h.m.$$$$\omega =\sqrt{\frac{(m+M)eg}{2I_P}}=\frac{1}{2}\sqrt{\frac{[4(M-m)L+3MR]g}{2m(\frac{L^2}{3}+R^2+RL)+\frac{13{MR}^2}{10}}}$$$$T=\frac{2\pi }{\omega }=4\pi \sqrt{\frac{20m(\frac{L^2}{3}+R^2+RL)+13{MR}^2}{10[4(M-m)L+3MR]g}}$$

Commented by Tawa11 last updated on 20/Aug/22

$$\mathrm{Great}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com