In-the-figure-shown-below-all-surfaces-are-smooth-strings-and-pulley-are-ideal-If-the-wedge-is-moving-with-acceleration-a-towards-the-right-then-the-acceleration-of-the-block-with-respect-to-the-w

Question Number 24301 by Tinkutara last updated on 15/Nov/17

In the figure shown below, all surfaces

are smooth, strings and pulley are ideal .

If the wedge is moving with acceleration

a towards the right, then the acceleration

of the block with respect to the wedge

at that instant is

Commented by Tinkutara last updated on 15/Nov/17

Commented by ajfour last updated on 15/Nov/17

l e t l e n g t h o f s t r i n g f r o m t o p o f

w e d g e t o b l o c k m b e y a n d f r o m

w a l l p u l l e y t o w e d g e b e x . T h e n

y + x + \sqrt{h^{2} + x^{2}} = l

\Rightarrow v_{r e l} + v_{M} + \frac{{x v}_{M}}{\sqrt{h^{2} + x^{2}}} = 0

o r a_{r e l} + a + \frac{(v_{M}^{2} + x a) \sqrt{h^{2} + x^{2}} - {x v}_{M} (\frac{{x v}_{M}}{\sqrt{h^{2} + x^{2}}})}{h^{2} + x^{2}} = 0

s o a_{r e l} = - a - \frac{(v_{M}^{2} + x a) (h^{2} + x^{2}) - {({x v}_{M})}^{2}}{(h^{2} + x^{2}) \sqrt{h^{2} + x^{2}}}

= - a - \frac{h^{2} (v_{M}^{2} + x a) + x^{3} a}{(h^{2} + x^{2}) \sqrt{h^{2} + x^{2}}}

= - a - \frac{h^{2} v_{M}^{2}}{{(h^{2} + x^{2})}^{3 / 2}} - \frac{x a}{\sqrt{h^{2} + x^{2}}}

i f o n l y s y s t e m h a s s t a r t e d n o w

f r o m r e s t a n d i n i t i a l r e l a t i o n

i s s o u g h t f o r, w e c a n h a v e

v_{M} = 0

a_{r e l} = - a (1 + \cos θ_{0})

t h e m i n u s s i g n i s t h e r e b e c a u s e

(\frac{d x}{d t}) i s - v e h e r e, a n d (\frac{d y}{d t}) i s + v e

i f w e d g e a c c e l e r a t e s t o w a r d s r i g h t .

s o (\frac{d^{2} x}{{d t}^{2}}) = a (i h a v e a s s u m e d)

b u t a c c o r d i n g t o q u e s t i o n i t

i s (\frac{d^{2} x}{{d t}^{2}}) = - a_{g i v e n i n q u e s t i o n} .

g i v e n a n s w e r m u s t b e

a_{r e l} = a (1 + \cos θ) .

Commented by Tinkutara last updated on 15/Nov/17

Thank you very much Sir!

Your solution is right .

Commented by Tinkutara last updated on 15/Nov/17