A cylinder of height h=1m has a narrow vertical slit along its wall running up to a length ℓ =40cm from the bottom. The width of the slit b=1mm. The cylinder is filled with water with the slit closed. If the force experienced by the vessel immediately after the slit is opened is (x/10)N , what is the value of x ?

Question

A cylinder of height h=1m has a narrow vertical slit along its wall running up to a length ℓ =40cm from the bottom. The width of the slit b=1mm. The cylinder is filled with water with the slit closed. If the force experienced by the vessel immediately after the slit is opened is (x/10)N , what is the value of x ? <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-ke0boksimrkbcmvm.jpg" /> (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Consider an element of the slit of length

d x

at a depth

x

from the top.
Velocity of efflux at

x = 2 gx .

Force on elementary area

=

rate of change of momentum

= (b d x ρ v) v = b ρ d x (2 gx)

∴ F

(total force experienced)

= 2 b ρ g \int_{h - l}^{h} x d x = b ρ g ℓ (2 h - ℓ)

or or

F = 1 0^{- 3} \times 1000 \times 10 \times 0.4 (2 - 0.4) = 6.40 N

Consider an element of the slit of length dx at a depth x from the top. Velocity of efflux at x=2gx​. Force on elementary area = rate of change of momentum =(bdxρv)v=bρdx(2gx) ∴F (total force experienced) =2bρg∫h−lh​xdx=bρgℓ(2h−ℓ) or or F=10−3×1000×10×0.4(2−0.4)=6.40N