Question-29322

Question Number 29322 by ajfour last updated on 07/Feb/18

Commented by ajfour last updated on 07/Feb/18

F i n d a c c e l e r a t i o n o f s p h e r e a s

a f u n c t i o n o f a n g l e θ .

S y s t e m i s r e l e a s e d a t θ = θ_{0} (< 90 °);

(A l l s u r f a c e s a r e f r i c t i o n l e s s) .

Commented by ajfour last updated on 07/Feb/18

P l e a s e s o l v e o r c h e c k m y s o l u t i o n

m r W S i r .

Commented by ajfour last updated on 07/Feb/18

Commented by ajfour last updated on 07/Feb/18

x = R \cot \frac{θ}{2}

v = - \frac{ω R}{2} cosec^{2} \frac{θ}{2}

\Rightarrow v^{2} = \frac{ω^{2} R^{2}}{4} cosec^{4} \frac{θ}{2} \dots . (i)

a = \frac{d v}{d t} = - \frac{α R}{2} cosec^{2} \frac{θ}{2}

+ \frac{ω^{2} R}{2} cosec^{2} \frac{θ}{2} \cot \frac{θ}{2} \dots (i i)

N \sin θ = m a \dots . (i i i)

N R \cot \frac{θ}{2} - \frac{M g L \cos θ}{2} = (\frac{{M L}^{2}}{3}) α

\Rightarrow α = \frac{3}{{M L}^{2}} (\frac{m a R \cot \frac{θ}{2}}{\sin θ} - \frac{M g L \cos θ}{2})

\dots . . (i v)

\frac{M g L}{2} (\sin θ_{0} - \sin θ) = \frac{1}{2} {m v}^{2} + \frac{1}{2} (\frac{{M L}^{2}}{3}) ω^{2}

u s i n g (i) :

\frac{M g L}{2} (\sin θ_{0} - \sin θ) = \frac{m ω^{2} R^{2}}{8} cosec^{4} \frac{θ}{2}

+ \frac{{M L}^{2} ω^{2}}{6}

ω^{2} = \frac{\frac{M g L}{2} (\sin θ_{0} - \sin θ)}{\frac{{M L}^{2}}{6} + \frac{{m R}^{2}}{8} cosec^{4} \frac{θ}{2}} \dots . (A)

U s i n g e q . (A), (i i i), (i v) i n e q . (i i)

a = \frac{R}{2} cosec^{2} \frac{θ}{2} (\frac{3}{{M L}^{2}}) (\frac{M g L \cos θ}{2} - \frac{m a R \cot \frac{θ}{2}}{\sin θ})

+ \frac{R}{2} cosec^{2} \frac{θ}{2} \cot \frac{θ}{2} [\frac{\frac{M g L}{2} (\sin θ_{0} - \sin θ)}{\frac{{M L}^{2}}{6} + \frac{{m R}^{2}}{8} cosec^{4} \frac{θ}{2}}]

a [1 + \frac{3 {m R}^{2} cosec^{2} \frac{θ}{2} \cot \frac{θ}{2}}{2 {M L}^{2} \sin θ}] =

\frac{R}{2} cosec^{2} \frac{θ}{2} [\frac{3 g \cos θ}{2 L} + \frac{\frac{M g L}{2} (\sin θ_{0} - \sin θ) \cot \frac{θ}{2}}{\frac{{M L}^{2}}{6} + \frac{{m R}^{2}}{8} cosec^{4} \frac{θ}{2}}] .

Commented by mrW2 last updated on 07/Feb/18

s o l u t i o n i s c o r r e c t s i r!

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