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# A solid cylinder of height h and mass m is floating in a liquid of density ρ as shown in the figure. Find the acceleration of the vessel (in m/s2) containing liquid for which the relative downward acceleration of the completely immersed cylinder to one-third of that of the vessel. (Take g=10m/s2)

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Solution

## The weight of the cylinder must be balanced by buoyancy force in upward direction.ρ′Ahg=ρA34hg⇒ρ′=34ρNow assume vessel is moving upward with an acceleration of a1m/s2.Given the relative acceleration arel of cylinder is 1/3rd of vessel in downward direction.applying pseudo force ma1 in downward direction Writing equation of motion for cylinder mg+ma1−B1=macylinder...(i)where B1=(P2−P1)A and acylinder=a13mg+ma1−(P2−P1)A=ma13...(ii)now taking an imaginary cylinder of water mass having same dimensions of cylinder writing equation of motion for imaginary cylinder B2−mliquidg=mliquida1...(iii) where B2=(P2−P1)A(P2−P1)A−mliquidg=mliquida1Substituting value of (P2−P1)A in equation (ii)mg+ma1−mliquid(g+a1)=ma13...(iv)The ratio mmliquid=ρ′AhgρAhg=34 substituting this value in equation (iv)34g+34a1−(g+a1)=34a13...(v)a12=−g4=−5m/s2i.e. the cylinder should move in downward direction with acceleration of 5m/s2

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