A cylinder that has a 38.0-cm radius and is 50.0 cm deep is filled with air at 1

Question

A cylinder that has a 38.0-cm radius and is 50.0 cm deep is filled with air at 17.5°C and 1.00 atm shown in figure (a). A 16.0-kg piston is now lowered into the cylinder, compressing the air trapped inside as it takes equilibrium height hi as shown in figure (b). Finally, a 28.5-kg dog stands on the piston, further compressing the air, which remains at 17.5°C as shown in figure (c).

(a) How far down (h) does the piston move when the dog steps onto it?
__________ mm

(b) To what temperature should the gas be warmed to raise the piston and the dog back to hi?
__________ °C

Explanation / Answer

A = area of cross section of cylinder = 3.14(0.38m)2 = 0.453m2

a) Assuming air to be ideal gas we have the formula:

PV = nRT

now, the temperature remains constant, So we have,

P1V1 = P2V2

Where P1 = pressure in the cylinder when there is no piston

P2 = pressure in the cylinder with piston

V1 = initial volume of air in the cylinder when there is no piston

V2 = volume in the cylinder with piston

when piston is lowered, pressure is

P2 = 1atm + Mg/A (M = mass of piston, A area of cylinder/piston)

or P2 = 101000N/m2 + 16kg(9.8m/s2)/[0.453m2] = 101346.13N/m2

P1 = 1atm = 101000N/m2

V1 = Ah (where h = 50cm) = (0.453m2)(0.5m) = 0.2265m3

V2 = Ahi

So using the formula

P1V1 = P2V2, we have

(1atm)Ah = (1atm + Mg/A)(Ahi)

or P1h = P2hi

or hi = P1h/P2 = (101000N/m2)(0.5m)/(101346.13N/m2)

or hi = 0.498m

Now let us suppose after the dog stands on piston

pressure be P3 = P2 + Mdg/A, where Md is mass of the dog

or P3 =  101346.13N/m2 + (28.5kg)(9.8m/s2)/(0.453m2) = 101962.686N/m2

Volume after the dog stands on piston is V3 = A(hi - h)

Using the formula P2V2 = P3V3

we get P2Ahi = P3A(hi - h)

or  P2hi = P3(hi - h)

or P2/P3 = 1 - h/hi

or h/hi = 1 - P2/P3 = 1 - (101346.13N/m2)/(101962.686N/m2)

h = 6mm.

So, the piston moves h = 6mm when the dog steps onto it.

b) Here volume changes from V3 = A(hi - h) to V2 = Ahi, pressure remains same as P3, Temperature raised from T3 = 17.50C to T.

using the ideal gas formula we have

V3/T3 = V2/T

A(hi - h)/T3 = Ahi/T

(hi - h)/T3 = hi/T

T = hiT3/(hi - h) = (0.498m)(290.5)/(0.498 - 0.006)

T = 294.04K = 21.040C

So the gas should be warmed to temperature T = 21.040C to raise the piston and the dog back to hi.

This concludes the answers. Check the answer and let me know if it's correct. If you need anymore clarification or correction I will be happy to oblige....

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