Question-53593

Question Number 53593 by ajfour last updated on 23/Jan/19

Commented by ajfour last updated on 23/Jan/19

H a v i n g r e m o v e d t h e l a r g e s t c u b e

f r o m a h e m i s p h e r e, i t c a n r e s t i n

e q u i l i b r i u m o n a r o u g h i n c l i n e

p l a n e o f i n c l i n a t i o n α, i n t h e

m a n n e r s h o w n . F i n d θ .

(s o u r c e : S t a t i c s & D y n a m i c s

b y S . L . L o n e y)

A n s w e r g i v e n : θ = \sin^{- 1} [\frac{8 (3 π - \sqrt{6})}{9 π - 8} \sin α] .

Answered by mr W last updated on 23/Jan/19

Commented by mr W last updated on 23/Jan/19

R = r a d i u s o f s p h e r e

a^{2} + a^{2} + {(2 a)}^{2} = R^{2}

\Rightarrow a = \frac{R}{\sqrt{6}}

M = m a s s o f h e m i s p h e r e

m = m a s s o f c u b e

M = \frac{2 π R^{3} ρ}{3}

m = 8 a^{3} ρ = \frac{4 R^{3} ρ}{3 \sqrt{6}} = \frac{2 M}{π \sqrt{6}}

c e n t e r o f m a s s o f h e m i s p h e r e :

b = \frac{3 R}{8}

c e n t e r o f m a s s o f h e m i s p h e r e w i t h h o l e :

c = \frac{M b - m a}{M - m} = \frac{\frac{M 3 R}{8} - \frac{2 M}{π \sqrt{6}} \times \frac{R}{\sqrt{6}}}{M - \frac{2 M}{π \sqrt{6}}} = \frac{R (9 π - 8)}{8 (3 π - \sqrt{6})}

\frac{c}{\sin α} = \frac{R}{\sin (π - θ)}

\Rightarrow \sin θ = \frac{R}{c} \sin α = \frac{8 (3 π - \sqrt{6})}{9 π - 8} \times \sin α

\Rightarrow θ = \sin^{- 1} [\frac{8 (3 π - \sqrt{6})}{9 π - 8} \times \sin α]

Commented by ajfour last updated on 23/Jan/19

W o n d e r f u l q u e s t i o n a n d y o u r s o l u t i o n,

S i r!

Commented by mr W last updated on 23/Jan/19

t h a n k y o u s i r!

i n m y s o l u t i o n t h e s i d e l e n g t h o f c u b e

i s 2 a, t h e r e f o r e i g o t a = \frac{R}{\sqrt{6}} .

Commented by ajfour last updated on 23/Jan/19

t h a n k s_{∣} \underset{\underset{⌣}{-}}{\overset{\overset{⌢}{⌢}}{. ._{∣}}}) S i r .

Answered by ajfour last updated on 23/Jan/19

Commented by mr W last updated on 23/Jan/19

n i c e d i a g r a m!

i {d i d n}^{'} t m a k e s u c h a d i a g r a m, s o i

m a d e a m i s t a k e a t f i r s t .

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