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Q.

A uniform rectangular block of height h whose base is a square of side a rests on a rough horizontal plane. The plane is gradually tilted about a line parallel to two edges of the base then, (where   is the coefficient of friction)
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a

The block will simultaneously topple and slide if a =μh 

b

The block is just on point of toppling when  tanα<ah

c

The block slides fast before toppling a >μh   and the block will topple first       before  sliding if a <μh   

d

The block is just on point of toppling when  tanα=ah

answer is C.

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Detailed Solution

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Let rectangle OPQR be the vertical section of the block through its centre of gravity G, where
OP = a and PQ = h
When the block is just on the point of toppling over, let  be the inclination of the plane to the horizontal and in this case the vertical line through G must pass through O. from G draw GL perpendicular to OP. 
Then in ΔOGL , we have
  tanα=OLGL=12a12h=ah................(i)
When the block is on the point of sliding, the inclination of the plane to the horizontal must
 be λ , given by 
  μ=tanλ
  The block will slide or topple over according as  
i.e.,   tanλ<or>tanα
i.e.,  μ<or>a/h
i.e.,  μh<or>a
   a>or<μh
 

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A uniform rectangular block of height h whose base is a square of side a rests on a rough horizontal plane. The plane is gradually tilted about a line parallel to two edges of the base then, (where   is the coefficient of friction)