A glass tube has several different cross-sectional areas with the values indicat

Question

A glass tube has several different cross-sectional areas with the values indicated in the figure. A piston at the left end of the tube exerts pressure so that the mercury sample within the tube flows from the right end into the atmosphere with a speed of 8.00m/s. Three points within the tube are labeled A, B, and C. Atmospheric pressure is 1.01 x 10^5 N/m^2 and the density of mercury is 13,600 kg/m^3. (a) What is the speed of the mercury at point A? (b) What is the force applied to the piston? (c) Determine the height h of mercury in the manometer with the evacuated upper end.

Explanation / Answer

a) Apply Continuty equation

A_A*v_A = A_C*v_C

==> v_A = v_C*(A_C/A_A)

= 8*(6/12)

= 4 m/s <<<<<<<-------------Answer

b) Apply Bernoulli's principle

P_A + 0.5*rho*v_A^2 = P_atm + 0.5*rho*v_C^2

P_A = P_atm + 0.5*rho*(v_C^2 - v_A^2)

= 1.01*10^5 + 0.5*13600*(8^2 - 4^2)

= 427400 N/m^2

P_A = F/A_A

==> F = P_A*A_A

= 427400*12*10^-4

= 512.88 N <<<<<<<----------------Answer

C) Apply Continuty equation

A_B*v_B = A_C*v_C

==> v_A = v_C*(A_C/A_B)

= 8*(6/5.6)

= 8.57 m/s

Apply Bernoulli's principle

P_B + 0.5*rho*v_B^2 = P_atm + 0.5*rho*v_C^2

P_B = P_atm + 0.5*rho*(v_C^2 - v_B^2)

= 1.01*10^5 + 0.5*13600*(8^2 - 8.57^2)

= 36775 pa

P_B = rho*g*h

==> h = P_B/(rho*g)

= 36775/(13600*9.8)

= 0.276 m <<<<<<<-------------Answer

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