A cylinder that has a 445cm radius and is 500 cm deep is fil

A cylinder that has a 44.5-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 21.0-kg piston is now lowered into the cylinder, compressing the air trapped inside as it takes equilibrium height h_i as shown in figure (b). Finally, a 25.5-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 h_i?
____________ °C

Solution

a)

With Piston alone :T=Constant ,so

PV = P_oV_o

P(Ah_i)=P_o(Ah_f)

Since Area A=constant

=>P=P_o(h_o/h_i)

since P=P_o+(m_pg/A)

=>P_o+(m_pg/A)=P_o(h_o/h_i)

=>h_i=h_o/[1+(m_pg/P_oA)]

h_i =0.5/[1+(21*9.8/(1.01325*10⁵)(pi*0.445²))]

h_i=49.837 cm

With Dog standing on the piston

h\' =0.5/[1+((21+25.5)*9.8/(1.01325*10⁵)(pi*0.445²))]

h\'=49.64 cm

So Piston moves downwards by

h =h_i-h\' =49.837-49.64 =0.196 cm

h=1.96 mm

b)

With Pressure constant

=>V/T =V\'/T\'

Ah_i/T =Ah\'/T_i

=>T=T_i(h_i/h\') =(273.2+17.5)*(49.837/49.64)

T=291.86 K

T=291.86-273.2 =18.66^oC