The-system-is-released-from-rest-All-surfaces-are-smooth-Find-the-angle-at-which-the-acceleration-of-wedge-is-maximum-given-M-m-1-2-

Question Number 23617 by Tinkutara last updated on 02/Nov/17

The system is released from rest . All

surfaces are smooth . Find the angle θ at

which the acceleration of wedge is

maximum . (given \frac{M}{m} = \frac{1}{2})

Commented by Tinkutara last updated on 02/Nov/17

Commented by ajfour last updated on 02/Nov/17

m g \cos θ - N = m A \sin θ \dots (i)

(A b e i n g a c c . o f w e d g e)

a n d N \sin θ = M A \dots . (i i)

s o, A = \frac{m g \sin θ \cos θ}{M + m \sin^{2} θ}

g i v e n m = 2 M, t h e r e f o r e

A = \frac{2 g \sin θ \cos θ}{1 + 2 \sin^{2} θ} = \frac{2 g \tan θ}{1 + 3 \tan^{2} θ}

= \frac{2 g}{(\frac{1}{\tan θ} + 3 \tan θ)}

F o r m a x i m u m A, \frac{1}{\tan θ} = 3 \tan θ

\Rightarrow \tan θ = \frac{1}{\sqrt{3}} o r θ = 30 ° .

Commented by Tinkutara last updated on 03/Nov/17

Thank you very much Sir!

Answered by Tinkutara last updated on 03/Nov/17

Commented by Tinkutara last updated on 03/Nov/17

Commented by Tinkutara last updated on 03/Nov/17

Commented by Physics lover last updated on 03/Nov/17

H o w d i d M r A j f o u r f i n d t h e

m i n i m u m v a l u e o f t h a t t r i g o n o m e t r i c

e q u a t i o n . ?

[C o t θ + 3 t a n θ]

Answered by Physics lover last updated on 02/Nov/17

W h a t a b o u t t h i s o n e :

a_{w e d g e} = \frac{(m g \times S i n θ \times C o s θ)}{M + m}

a s t h e r e i s n o f o r c e i n h o r i z o n t a l

d i r e c t i o n \Rightarrow h o r i z o n t a l a c c e l e r a t u o n

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

z e r o . A n d w e c a n u s e t h e a b o v e

f o r m u l a .

{[a_{w e d g e}]}_{m a x} \Rightarrow {[S i n θ C o s θ]}_{m a x}

\Rightarrow θ = 45 °

? ? ? ?

w h a t i s w r o n g i n t h i s o n e ?

Commented by ajfour last updated on 02/Nov/17

h o w m g \sin θ \cos θ ?

Commented by ajfour last updated on 02/Nov/17

w h a t w o u l d y o u s a y f o r a c c .

o f b l o c k, w h a t i s i t, a t w h a t

a n g l e w i t h h o r i z o n t a l ?

a c c . o f b l o c k i s n o t m g \sin θ

d o w n t h e i n c l i n e d i r e c t i o n

f r o m g r o u n d, i f o b s e r v e r o b s e r v e s

m o t i o n o f b l o c k a n d w e d g e,

b o t h f r o m g r o u n d f r a m e .

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