A modern-day zeppelin holds 9,440 m 3 of helium. Compute its maximum payload at

Question

A modern-day zeppelin holds 9,440 m3 of helium. Compute its maximum payload at sea level. (Assume the helium and air to be at 0°C and 1 atm.)

Problem Statement: A modern-day zeppelin holds 7,800 m3 of helium. Compute its maximum payload at sea level. (Assume the helium and air to be at 0°C and 1 atm.)

Step 1: Write down what you know in symbolic form:
For this problem we are given the volume in cubic meters, therefore we will use the fact that the weight density of a substance can be represented as the mass density of the substance times the acceleration due to gravity: Dw = Dg (see page 146 in the text).
Volume: V = 7800 m3
Mass Density of air: D(air) = 1.29 kg/m3
Mass Density of helium: D(helium) = 0.18 kg/m3

Step 2: Write down what you don't know in symbolic form:
Weight of helium: W = ?
Buoyant force: Fb = ?
Maximum payload or Net Force: Fnet = ?

Step 3: Find an equation that contains what you know and what you don't know:
Weight of helium: W = D(helium) x g x V(of helium)
Buoyant force: Fb = D(air) x g x V(air)
In this case V(of helium) = V(air) = V
Net Force: Fnet = Fb - W

Step 4: Solve the equation for what you don't know:
Net Force: Fnet = Fb - W = D(air) x g x V - D(helium) x g x V = [D(air) - D(helium)] x g xV

Step 5: Plug in (substitute) what you know and you will find out what you don't know:
Net Force: Fnet = Fb - W = [D(air) - D(helium)] x g x V = [1.29 kg/m3 - 0.18 kg/m3 ] x 9.8 m/s2 x 7,800 m3 = 1.11 kg/m3 x 9.8 m/s2 x 7,800 m3 = 84,848. 4 kg m/s2 (m3/m3) = 84,848.4 N

Explanation / Answer

Step 1: Write down what you know in symbolic form:
For this problem we are given the volume in cubic meters, therefore we will use the fact that the weight density of a substance can be represented as the mass density of the substance times the acceleration due to gravity: Dw = Dg
Volume: V = 7800 m3
Mass Density of air: D(air) = 1.29 kg/m3
Mass Density of helium: D(helium) = 0.18 kg/m3

Step 2: Write down what you don't know in symbolic form:
Weight of helium: W = ?
Buoyant force: Fb = ?
Maximum payload or Net Force: Fnet = ?

Step 3: Find an equation that contains what you know and what you don't know:
Weight of helium: W = D(helium) x g x V(of helium) = 0.18 kg/m3 * 9.81*7800 = 13773.24 N
Buoyant force: Fb = D(air) x g x V(air) = 1.29*9.81*7800 =98708.22 N
In this case V(of helium) = V(air) = V
Net Force: Fnet = Fb - W = 98708.22 -13773.24 =84934.98 N

Step 4: Solve the equation for what you don't know:
Net Force: Fnet = Fb - W = D(air) x g x V - D(helium) x g x V = [D(air) - D(helium)] x g xV

Step 5: Plug in (substitute) what you know and you will find out what you don't know:
Net Force: Fnet = Fb - W = [D(air) - D(helium)] x g x V = [1.29 kg/m3 - 0.18 kg/m3 ] x 9.8 m/s2 x 7,800 m3 = 1.11 kg/m3 x 9.8 m/s2 x 7,800 m3 = 84,848. 4 kg m/s2 (m3/m3) = 84,848.4 N

similarly we can calculate for 9440 m3 volume

Weight of helium: W = D(helium) x g x V(of helium) = 0.18 kg/m3 * 9.81*9440 = 16669.152 N
Buoyant force: Fb = D(air) x g x V(air) = 1.29*9.81*9440 =119462.25 N
In this case V(of helium) = V(air) = V
Net Force: Fnet = Fb - W = 119462.25 -16669.152 =102793.104 N.........Ans.

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