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Water of density p flows with a linear s...

Water of density `p` flows with a linear speed `v` through a horizontal rubber tube having the form of a ring of radius `R`. If the diameter of the tube is `d(ltlt R)`, find the tension in the rubber tube.

Text Solution

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Consider a small portion of the tube as shown in Fig. 7.68(b) . Centripetal force is provided by radial component of tension in the tube . So , if T is the tension in the tube

F = 2T sin `theta`
So that `(v^(2)"dm")/(R) = 2 T "sin" theta` , i.e., `T = (v^(2) "dm")/(2 R "sin" theta) " " ..... (i)`
But dm = `pi (d//2)^(2) (2R theta)rho ` [ as m = `pi r^(2)//rho) " " .... (ii)`
Substituting the value of dm from Eqn. (ii) in (i) , we get
`T = (pi rho v^(2) theta d^(2))/(4 "sin" theta) = (1)/(4) pi rho v^(2)d^(2)[ "as" underset(theta to 0)("lim") ("sin" theta)/(theta) = 1]`
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