Equipment: latex balloon, two syringes of $10~\mathrm{ml}$ (one without « ears »), IV drip tube, a cup of water, thread, plastic ruler, paper towels.

The excess pressure inside a rubber balloon reaches several $\mathrm{kPa}$ and this pressure is determined by elastic properties of latex. In the problem, you are asked to investigate this dependence using a DIY liquid manometer and determine the thickness of balloon's wall.

Density of water $\rho=1.00~\mathrm{g}/\mathrm{cm}^3$, acceleration due to gravity $g=9.8~\mathrm{m}/\mathrm{s}^2$. Throughout the entire task, consider that shape of balloon in a sphere!

Ensure that there are no air bubbles in the tube during pressure measurements.

Attention: you have only one attempt to perform the experiment. If you inflate the balloon again, its properties will change, and the measurements will have to be done from the begining.

To measure the perimeter of an object with a complex shape, you can use a thread wrapped around it. If the length of the thread exceeds $30~\mathrm{cm}$, you can carefully fold it in half and measure half of thread's length using a ruler.

Part A. Warm-up (1.6 point)

A1  ^0.30 Determine the inner diameter $D_\mathrm{in}$ of the syringe. Sketch your setup.

A2  ^0.70 Determine the outer diameter $D_\mathrm{outer}$ of the syringe.

A3  ^0.30 Determine the thickness $t$ of syringe's wall. Sketch your setup.

A4  ^0.30 Determine the inner diameter $d_\mathrm{in}$ of the tube. Sketch your setup.

Part B. Main experiment (8.4 points)

B1  ^2.80 Gradually deflate the balloon by pulling its edge and measure the dependence of the level difference $h$ on the balloon's radius $R$.

Perform $3$ measurements for $R>10~\mathrm{cm}$, 10 measurments for $4.0~\mathrm{cm} < R \leq 10~\mathrm{cm}$, and 3 measurments for $R \leq 4.0~\mathrm{cm}$.

Keep in mind that the balloon deflate faster when it is smaller.

B2  ^1.80 Plot the graph of the excess pressure $\Delta p$ vs the radius of the balloon $R$.

B3  ^0.10 What is the maximum value of excess presure, $\Delta p_\mathrm{max}$, reached in the balloon during the measurements?

Consider a small « square » element with sides $a \times a$ on the surface of a thin-walled, elastic sphere of initial radius $R_0$ and wall thickness $\delta _0$. Suppose the sphere is stretched by some forces such that its radius increases by a factor of $\lambda$, becoming $\lambda R_0$. Consequently, the sides of the small « square » element also increases to $\lambda a \times \lambda a$.

Under these conditions, an elastic force acts on each side of deformed « square ». The magnitude of this force is given by
\[ F = (\lambda - 1) a \delta E ,\]where $E$ is the Young's modulus of the material, and $\delta$ is the wall thickness of the deformed sphere.

B4  ^0.20 Determine the $\delta$ if the volume of the sphere's walls remains constant during the stretching. Express your answer in terms of $\lambda$ and $\delta_0$.

Β5  ^2.20 Plot the graph of $\Delta p \cdot R^3$ vs $R$. Determine the $\mathrm{slope}$ and $\mathrm{offset}$ of the linear fit to the data.

B6  ^0.20 What is the radius $R_0$ of the unstretched balloon?

The Young's modulus of the latex is known to be $E=2.0~\mathrm{MPa}$. Let's apply the theoretical model of a thin-walled, ellastic sphere to the ballon.

B7  ^1.10 Using the graph plotted in question B6, determine the thickness $\delta_0$ of the unstretched balloon's wall.