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# 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 dl of the cork along the periphery (see the figure). Assuming that the temperature of the gas is uniform at any instant, calculate dN/dl.

Answer :

Given

Pressure of gas P_{1}=1atm=10^{5}Pa

Radius of tube R=5cm=0.05m

So, area of tube = (0.05)^{2}

Length of tube =20cm=0.2m

So, volume=arealength

Volume of cylindrical tube = (0.05)^{2}0.2=0.0016m^{2}

Initial temperature T_{1} =300K

Final temperature T_{2}=600K

Coefficient of friction =0.2

Let final pressure be P_{2}. So, volume of the gas remains same until pressure becomes P_{2} and then corks pop out. Number of moles will also be same. So, we can apply ideal gas equation which is

PV=nRT

Where V= volume of gas

R=gas constant =8.3JK^{-1}mol^{-1}

T=temperature

n=number of moles of gas

P=pressure of gas.

Net pressure on cork = P_{2}-P_{1}=210^{5} - 10^{5}=10^{5}Pa

We know that

So, force acting on the cork=pressure on corkarea of cork

F=10^{5}(0.05)^{2}

According to law of friction,

F=N

Where F=force of friction

N=normal to the surface of cork

=coffieicent of friction

Now, friction is always equal to the applied force until body starts to slide.

So,

In question N is denoted as dN. So, N=dN

And dl=length of cork around periphery of cork i.e. dl=circumference of cork.

Thus, the value of .

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