Six identical uniform rods $P Q, Q R, R S, S T, T U$ and $U P$ each weighing W are freely joined at their ends to form a hexagon. The rod $P Q$ is fixed in a horizontal position and middle points of $P Q$ and $S T$ are connected by a vertical string. The tension in string is

W

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3 W

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2 W

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4 W

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Solution

The correct option is D $3 W$
Let a small displacement be given to the system in vertical plane in frame such that ST remains horizontal then let vertical displacement of centres of rods UP and QR be y then vertical displacement of centres of UT and RS will be 3y and that of TS will by 4y. Equating the total virtual work to zero we get.
$(W + W) δ y + (W + W) 3 δ y + W (4 δ y) - T (4 δ y) = 0$
(where T is the tension in thread)
$\Rightarrow 2 W + 6 W + 4 W = 4 T$
$\Rightarrow T = 3 W$

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Six identical uniform rods PQ,QR,RS,ST,TU and UP each weighing W are freely joined at their ends to form a hexagon. The rod PQ is fixed in a horizontal position and middle points of PQ and ST are connected by a vertical string. The tension in string is

Six identical uniform rods $P Q, Q R, R S, S T, T U$ and $U P$ each weighing W are freely joined at their ends to form a hexagon. The rod $P Q$ is fixed in a horizontal position and middle points of $P Q$ and $S T$ are connected by a vertical string. The tension in string is