A spherical rubber balloon has an initial thickness of 0.45mm, and diameter 250mm. Final diameter of 1000mm.
Find final thickness, true stress in the plane of the balloon wall, Air pressure required.
(Assume rubber is idea and obeys Guassian statistics, and take the shear modulus to be 1MPa.)
To find the final thickness of the rubber balloon, we can use the principle of conservation of volume.
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
Given that the initial diameter of the balloon is 250mm, the initial radius (r1) can be calculated as:
r1 = 250mm / 2 = 125mm = 0.125m
The final diameter of the balloon is 1000mm, so the final radius (r2) can be calculated as:
r2 = 1000mm / 2 = 500mm = 0.5m
Using the conservation of volume, we have:
V1 = V2
Substituting the formulas for the volumes of the spheres:
(4/3) * π * r1^3 = (4/3) * π * r2^3
Simplifying:
r1^3 = r2^3
(0.125m)^3 = (0.5m)^3
0.001953125 = 0.125
Therefore, the final thickness of the rubber balloon is the initial thickness multiplied by the cube root of the ratio of the initial radius to the final radius:
Final thickness = 0.45mm * (0.125/0.5)^(1/3)
Final thickness ≈ 0.45mm * (0.25)^(1/3)
Final thickness ≈ 0.45mm * 0.6299
Final thickness ≈ 0.282 mm
The final thickness of the rubber balloon is approximately 0.282 mm.
To find the true stress in the plane of the balloon wall, we can use the formula:
True stress = Shear stress * (1 + (2 * Shear strain))
Given that the shear modulus is 1 MPa, the true stress can be calculated as:
True stress = 1 MPa * (1 + (2 * Shear strain))
To find the shear strain, we can use the formula:
Shear strain = (change in thickness) / (initial thickness)
For the rubber balloon, the change in thickness is the final thickness minus the initial thickness:
Change in thickness = Final thickness - Initial thickness
Substituting the given values:
Change in thickness = 0.282mm - 0.45mm
Change in thickness = -0.168mm
The shear strain can be calculated as:
Shear strain = (-0.168mm) / (0.45mm)
Shear strain ≈ -0.373
Substituting the shear strain into the formula for true stress:
True stress = 1 MPa * (1 + (2 * (-0.373)))
True stress ≈ 1 MPa * (1 + (-0.746))
True stress ≈ 1 MPa * 0.254
True stress ≈ 0.254 MPa
The true stress in the plane of the balloon wall is approximately 0.254 MPa.
To find the air pressure required, we can use the formula for pressure:
Pressure = Force / Area
The force can be calculated using the formula:
Force = True stress * Area
Since the balloon is spherical, the area can be calculated using the formula:
Area = 4 * π * r^2
Substituting the values:
Area = 4 * π * (0.5m)^2
Area = 4 * π * 0.25m^2
Area = π m^2
Substituting the area into the formula for force:
Force = 0.254 MPa * π m^2
Force = 0.254π MPa * m^2
Substituting the force into the formula for pressure:
Pressure = (0.254π MPa * m^2) / π m^2
Pressure = 0.254 MPa
Therefore, the air pressure required is approximately 0.254 MPa.
To find the final thickness of the balloon, we can use the principle of volume conservation. The volume of the balloon remains constant throughout the process.
The initial volume of the balloon can be calculated using the formula for the volume of a sphere:
V_initial = (4/3) * π * (radius_initial)^3
= (4/3) * π * (diameter_initial/2)^3
And the final volume of the balloon can be calculated in the same way:
V_final = (4/3) * π * (radius_final)^3
= (4/3) * π * (diameter_final/2)^3
Since the volume is conserved, we can equate the initial and final volumes and solve for the final thickness:
V_initial = V_final
(4/3) * π * (diameter_initial/2)^3 = (4/3) * π * (diameter_final/2)^3
We are given the initial diameter (250mm) and the final diameter (1000mm). Substituting these values into the equation:
(4/3) * π * (250/2)^3 = (4/3) * π * (1000/2)^3
Simplifying the equation:
(125/2)^3 = (500/2)^3
(125/2)^3 = (125)^3
∴ (125/2)^3 = 125^3/2^3
(125/2)^3 = 125^3/8
Taking the cube root of both sides:
125/2 = 125/2∛8
125/2 = 125/2∛(2^3)
125/2 = 125/2∛2^3
125/2 = 125/2 * 2^(3/3)
125/2 = 125/2 * 2
Therefore, the final thickness of the balloon is 0.45mm.
To calculate the true stress in the plane of the balloon wall, we can use the formula:
True Stress = Applied force / Area
For a spherical balloon, the force acting on the plane of the balloon wall is the pressure difference between the inside and outside of the balloon, multiplied by the surface area of the balloon wall. The surface area of a sphere can be calculated using the formula:
Surface Area = 4 * π * (radius^2)
The pressure difference required to achieve a certain diameter change can be calculated using the equation:
Pressure difference = (Shear modulus * Initial thickness) / (Final diameter - Initial diameter)
We are given the initial thickness (0.45mm), the shear modulus (1MPa), and the final and initial diameters (1000mm and 250mm).
Substituting these values into the formula, we can calculate the pressure difference required:
Pressure difference = (1 * 0.45) / (1000 - 250)
= 0.45 / 750
= 0.0006 MPa
Now, substituting this pressure difference and the initial thickness into the formula for true stress:
True Stress = (Pressure difference * Initial thickness) / (Final diameter - Initial diameter)
= (0.0006 * 0.45) / (1000 - 250)
= 0.00027 / 750
= 3.6e-7 MPa (approximately)
Therefore, the final thickness of the balloon is 0.45 mm, the true stress in the plane of the balloon wall is approximately 3.6e-7 MPa, and the air pressure required is 0.0006 MPa.