Answer to Question #133189 in Physics for Cameron

Question

A 15 gram rubber balloon is filled with 18 L of Helium (density 0.179 kg/m3) and tied with a string to a child's wrist. Find the tension in the cord.

Answer 1

The free body diagram is shown on the figure. Here:

$\mathbf{F}_b = \rho_{air}\cdot g\cdot V_{He}$ is the buoyant force, $T$ is the tension in the cord and $\mathbf{mg}$ is the gravity force.

In projections on the vertical axis get:

$$F_b - mg - T = 0\\ T = F_b - mg = \rho_{air}\cdot g\cdot V_{He} - mg =g ( \rho_{air}\cdot V_{He} - m)$$

Let's substitute the numerical values: $\rho_{air} = 1.2 kg/m^3$, $V_{He} = 18L = 0.018 m^3$ and $g = 9.81m/s^2$.

Here $m$ is the mass of the ballon together with 18 L of Helium:

$$m = 0.015kg + V_{He}\cdot \rho_{He} = 0.015+ 0.018\cdot 0.179 = 0.018222\space kg$$

Finally, obtain:

$$T =g ( \rho_{air}\cdot V_{He} - m) =9.81\cdot ( 1.2\cdot 0.018 - 0.018222) \approx 0.033N$$

Answer. 0.033 N.