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 spherical rubber balloon, we can use the concept of volume preservation. According to the assumption of Gaussian statistics, when the balloon is inflated, the volume remains constant.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
We can use this formula to calculate the initial and final volumes of the balloon.
Initial volume (V1):
V1 = (4/3) * π * (r1)^3
= (4/3) * π * (125mm)^3 [since the diameter is given, we need to convert it to radius]
Final volume (V2):
V2 = (4/3) * π * (r2)^3
= (4/3) * π * (500mm)^3
Since the volume is preserved, V1 = V2. Therefore, we can equate the two expressions for volumes and solve for r2 (final radius).
(4/3) * π * (125mm)^3 = (4/3) * π * (500mm)^3
(125)^3 = (500)^3
125mm = 500mm
This means that the final radius of the balloon remains the same as the initial radius of 125mm.
Now that we have the final and initial radii, we can find the final thickness.
Final thickness (t2) = Final radius (r2) - Initial radius (r1)
= 500mm - 125mm
= 375mm
Next, let's calculate the true stress in the plane of the balloon wall.
True stress is defined as the force per unit area. In this case, since the balloon is inflated, the force acting on the balloon is the air pressure.
The force acting on the balloon wall is given by:
Force = Pressure * Area
The area is the surface area of the sphere, which can be calculated using the formula:
A = 4 * π * r^2
The initial area (A1) is:
A1 = 4 * π * (r1)^2
= 4 * π * (125mm)^2
The final area (A2) is:
A2 = 4 * π * (r2)^2
= 4 * π * (500mm)^2
Since we assume the rubber is ideal, the volume of the balloon is preserved during inflation, meaning the surface area remains constant. Therefore, A1 = A2.
Now, let's calculate the air pressure required.
Pressure2 = Force / A2
= Force / (4 * π * (500mm)^2)
However, we don't know the force acting on the balloon. We need to calculate it by considering the stress applied to the rubber balloon wall.
Stress is defined as the force applied per unit area.
True stress = Force / A1
We can rearrange this equation to solve for the force:
Force = True stress * A1
Now we can substitute this value of force back into the equation for pressure:
Pressure2 = (True stress * A1) / (4 * π * (500mm)^2)
The shear modulus of rubber is given as 1MPa, which is the same as 1 N/mm^2.
Therefore, the true stress in the rubber balloon wall is 1 N/mm^2.
Plugging in the known values, we can calculate the air pressure required.