1.2 litre of water leaked out from a tank with a base area of 800 cm square? Find the decrease in the water level in the tank.
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Answer:
Area of the hole of the water tank=2mm
2
Height of filled waterh=80cm
Area of the cross-section of the tanksA=0.4m
2
Acceleration due to gravityg=9.8ms
−2
≈10ms
−2
Pressure at the open surface and at the hole is equal to the atmospheric pressure.
(a)Velocity of water
v=
2gh
=
2×10×0.80
=4m/s
(b)Velocity of water when the tank is half-filled and h is
2
80
=40cm
2gh
=
2×10×0.40
=
8
m/s
(c)Volume=Ah=Avdt
=A×
2gh
dt
=(2mm
2
)
2gh
dt
Volume of the tank=Ah=V(say)
⇒
dt
dV
=A
dt
dh
⇒a
1
v
1
=A
dt
dh
⇒2×10
−6
2gh
=0.4
dt
dh
⇒dh=5×10
−6
2gh
dt
(d)∵dh=5×10
−6
2gh
dt
∴
2gh
dh
=5×10
−6
dt
On integrating we get
5×10
−6
∫
0
t
dt=
28
1
∫
0.8
0.4
h
dh
⇒5×10
−6
×t=
28
1
×2[h
2
1
]
0.8
0.4
⇒t=
28
1
×2[(0.4)
2
1
−(0.8)
2
1
]×
5×10
−6
1
⇒t=
4.47
1
×2×
3.16
2
×
5×10
−6
1
×
3600
1
h
∴t=6.51h
Thus, the time required to leak half of the water out is 6.51hrs.
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