Advertisements
Advertisements
प्रश्न
Considering the pressure p to be proportional to the density, find the pressure p at a height h if the pressure on the surface of the earth is p0.
Advertisements
उत्तर
Let p0 be the density of air on the surface of the earth.
As per the question, pressure ∝ density
⇒ `p/p_0 = ρ/ρ_0`
⇒ `ρ = ρ_0/p_0 p`
∴ `dp = - (ρ_0g)/p_0 pdh` ......[∵ dp = – ρgdh]
⇒ `(dp)/p = - (ρ_0g)/p_0 dh`
⇒ `int_(p_0)^p (dp)/p = - (ρ_0 g)/p_0 int_0^h dh` .....`[(∵ at h = p, r = p_0),(and at h = h, p = p)]`
⇒ In `p/ρ_0 = - (ρ_0 g)/p_0 h`
By removing log, `p = p_0e (- (ρ_0gh)/p_0)`
APPEARS IN
संबंधित प्रश्न
A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit?
Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation? Explain.
During blood transfusion the needle is inserted in a vein where the gauge pressure is 2000 Pa. At what height must the blood container be placed so that blood may just enter the vein? [Use the density of whole blood from Table 10.1].
A barometer tube reads 76 cm of mercury. If the tube is gradually inclined keeping the open end immersed in the mercury reservoir, will the length of mercury column be 76 cm, more than 76 cm or less than 76 cm?
A satellite revolves round the earth. Air pressure inside the satellite is maintained at 76 cm of mercury. What will be the height of mercury column in a barometer tube 1 m long placed in the satellite?
The three vessels shown in the following figure have same base area. Equal volumes of a liquid are poured in the three vessels. The force on the base will be
Equal mass of three liquids are kept in three identical cylindrical vessels A, B and C. The densities are ρA, ρB, ρC with ρA < ρB < ρC. The force on the base will be
Shows in the following figure a siphon. The liquid shown is water. The pressure difference PB − PAbetween the points A and B is
A barometer kept in an elevator reads 76 cm when it is at rest. If the elevator goes up with increasing speed, the reading will be ______.
Figure shows a capillary tube of radius r dipped into water. If the atmospheric pressure is P0, the pressure at point A is
The surface of water in a water tank on the top of a house is 4 m above the tap level. Find the pressure of water at the tap when the tap is closed. Is it necessary to specify that the tap is closed?
The area of cross section of the wider tube shown in figure is 900 cm2. If the boy standing on the piston weighs 45 kg, find the difference in the levels of water in the two tubes.
The weight of an empty balloon on a spring balance is W1. The weight becomes W2when the balloon is filled with air. Let the weight of the air itself be w. Neglect the thickness of the balloon when it is filled with air. Also neglect the difference in the densities of air inside and outside the balloon.
(a) W2 = W1
(b) W2 = W1 + w
(c) W2 < W1 + w
(d) W2 > W1
Suppose the glass of the previous problem is covered by a jar and the air inside the jar is completely pumped out. (a) What will be the answers to the problem? (b) Show that the answers do not change if a glass of different shape is used provided the height, the bottom area and the volume are unchanged.
If water be used to construct a barometer, what would be the height of water column at standard atmospheric pressure (76 cm of mercury) ?
Water leaks out from an open tank through a hole of area 2 mm2 in the bottom. Suppose water is filled up to a height of 80 cm and the area of cross section of the tanks is 0.4 m2. The pressure at the open surface and at the hole are equal to the atmospheric pressure. Neglect the small velocity of the water near the open surface in the tank. (a) Find the initial speed of water coming out of the hole. (b) Find the speed of water coming out when half of water has leaked out. (c) Find the volume of eater leaked out using a time interval dt after the height remained is h. Thus find the decrease in height dh in terms of h and dt.
(d) From the result of park (c) find the time required for half of the water to leak out.
Pressure decreases as one ascends the atmosphere. If the density of air is ρ, what is the change in pressure dp over a differential height dh?
