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 balloon, we can use the principle of volume conservation. At the initial state, the volume of the balloon can be calculated using the formula for the volume of a sphere:

V_initial = (4/3)π(r_initial)^3

where r_initial is the initial radius, which can be found by dividing the initial diameter by 2:

r_initial = (250mm) / 2 = 125mm

Converting the radius to meters:

r_initial = 125mm / 1000 = 0.125m

Plugging this value into the equation for the initial volume:

V_initial = (4/3)π(0.125m)^3

Next, we can calculate the final radius, r_final, using the same method:

r_final = (1000mm) / 2 = 500mm / 1000 = 0.5m

Now, we can use the principle of volume conservation to find the final thickness. The final volume, V_final, is equal to the initial volume:

V_final = V_initial

(4/3)π(r_final)^3 = (4/3)π(0.125m)^3

To find the final thickness, t_final, we need to subtract twice the final radius from the initial diameter:

t_final = r_final - r_initial

t_final = 0.5m - 0.125m

Therefore, the final thickness of the balloon is:

t_final = 0.375m = 375mm

Now, to find the true stress in the plane of the balloon wall, we can use the formula:

True stress = (force / area) = (shear modulus * strain)

Since the rubber is assumed to be ideal and obey Gaussian statistics, the strain in a thin-walled spherical balloon can be approximated as:

strain = (initial radius - final radius) / (initial radius)

Plugging in the values:

strain = (0.125m - 0.5m) / (0.125m) = -3

Next, we substitute the given shear modulus:

shear modulus = 1MPa = 1 x 10^6 Pa

Using the formula for true stress, we have:

True stress = shear modulus * strain = (1 x 10^6 Pa) * (-3)

Finally, to find the air pressure required to inflate the balloon to the final diameter, we can use the formula for pressure in a thin-walled balloon:

Pressure = 2 * True stress / (final radius - initial radius)

Plugging in the values:

Pressure = 2 * (-3) * (1 x 10^6 Pa) / (0.5m - 0.125m)

Simplifying:

Pressure = -6 x 10^6 Pa / 0.375m = -16 x 10^6 Pa / 3

Therefore, the air pressure required to inflate the balloon to the final diameter is:

Pressure ≈ -5.33 x 10^6 Pa