Advertisements
Advertisements
प्रश्न
Consider a rectangular block of wood moving with a velocity v0 in a gas at temperature T and mass density ρ. Assume the velocity is along x-axis and the area of cross-section of the block perpendicular to v0 is A. Show that the drag force on the block is `4ρAv_0 sqrt((KT)/m)`, where m is the mass of the gas molecule.
Advertisements
उत्तर
Consider the diagram
Let n = number of molecules per unit volume
Vrms = rms speed of the gas molecules
When a block is moving with speed v0, relative speed of molecules w.r.t. front face = v + v0
Coming head-on, momentum transferred to block per collision = 2m (v + v0), where, m = mass of the molecule.
The number of collisions in time Δt = m(v + v0)2nAΔt, where, A = area of cross-section of block and factor of 1/2 appears due to particles moving towards the block.
∴ The momentum transferred in time Δt = m(v + v0)2nAΔt from the front surface.
Similarly, momentum transferred in time Δt = m(v – v0)2nAΔt ......(From the back surface)
∴ Net force (drag force) = mnA[(v + v0)2 – (v – v0)2] .....[From front]
= mnA (4vv0)
= (4 mnAv)v0
= (4ρAv)v0 ......(i)
Where we have assumed ρ = `(mn)/V = M/V`
If v = velocity along the x-axis
Then, we can write KE = `1/2 mv^2 = 1/2 K_BT`
⇒ `v = sqrt((K_BT)/m)` .....`[(K_B = "Boltzmann constant"),(KE = "Kinetic energy"),(T = "Temperature")]`
∴ From equation (i), Drag force = (4ρAv)v0 = `4ρA sqrt((K_BT)/m) v_0`.
APPEARS IN
संबंधित प्रश्न
Which of the following quantities is the same for all ideal gases at the same temperature?
(a) The kinetic energy of 1 mole
(b) The kinetic energy of 1 g
(c) The number of molecules in 1 mole
(d) The number of molecules in 1 g
A gas cylinder has walls that can bear a maximum pressure of 1.0 × 106 Pa. It contains a gas at 8.0 × 105 Pa and 300 K. The cylinder is steadily heated. Neglecting any change in the volume, calculate the temperature at which the cylinder will break.
The temperature and pressure at Simla are 15.0°C and 72.0 cm of mercury and at Kalka these are 35.0°C and 76.0 cm of mercury. Find the ratio of air density at Kalka to the air density at Simla.
Use R=8.314J K-1 mol-1
The average translational kinetic energy of air molecules is 0.040 eV (1 eV = 1.6 × 10−19J). Calculate the temperature of the air. Boltzmann constant k = 1.38 × 10−23 J K−1.
One mole of an ideal gas undergoes a process `P = (P_0)/(1+(V/V_0)^2` where `p_0` and `V_0` are constants . Find the temperature of the gas when `V=V_0` .
Figure shows a cylindrical tube of radius 5 cm and length 20 cm. It is closed by a tight-fitting cork. The friction coefficient between the cork and the tube is 0.20. The tube contains an ideal gas at a pressure of 1 atm and a temperature of 300 K. The tube is slowly heated and it is found that the cork pops out when the temperature reaches 600 K. Let dN denote the magnitude of the normal contact force exerted by a small length dlof the cork along the periphery (see the figure). Assuming that the temperature of the gas is uniform at any instant, calculate `(dN)/(dt)`.
A glass contains some water at room temperature 20°C. Refrigerated water is added to it slowly. when the temperature of the glass reaches 10°C, small droplets condense on the outer surface. Calculate the relative humidity in the room. The boiling point of water at a pressure of 17.5 mm of mercury is 20°C and at 8.9 mm of mercury it is 10°C.
If a = 0.72 and r = 0.24, then the value of tr is ______.
Answer in brief:
What will happen to the mean square speed of the molecules of a gas if the temperature of the gas increases?
At what temperature will oxygen molecules have same rms speed as helium molecules at S.T.P.? (Molecular masses of oxygen and helium are 32 and 4 respectively).
When a gas is heated, its temperature increases. Explain this phenomenon on the basis of the kinetic theory of gases.
Energy is emitted from a hole in an electric furnace at the rate of 20 W when the temperature of the furnace is 727°C. What is the area of the hole? (Take Stefan’s constant σ to be 5.7 × 10-8 Js-1 m-2K-4.)
Under which condition laws of Boyle, Charles, and Gay-Lussac are valid?
A metal cube of length 4 cm radiates heat at the rate of 10 J/s. Find its emissive power at a given temperature.
The average translational kinetic energy of gas molecules depends on ____________.
What is the microscopic origin of temperature?
The average K.E. of hydrogen molecules at 27° C is E. The average K.E. at 627° C is ____________.
A ring of mass m and radius r rotates about an axis passing through its centre and perpendicular to its plane with angular velocity `omega`. Its kinetic energy is ______.
23Ne decays to 23Na by negative beta emission. Mass of 23Ne is 22.994465 amu mass of 23Na is 22.989768 amu. The maximum kinetic energy of emitted electrons neglecting the kinetic energy of recoiling product nucleus is ______ MeV.
2000 calories of radiant heat is incident on a body. If the body absorbs 550 calories of heat, find the coefficient of emmission of the body.
