Serway & Jewett — Fluid Mechanics: Problem 33

A helium-filled balloon (whose envelope has a mass of \( m_b = 0.250 \text{ kg} \)) is tied to a uniform string of length \( \ell = 2.00 \text{ m} \) and mass \( m = 0.0500 \text{ kg} \). The balloon is spherical with a radius of \( r = 0.400 \text{ m} \). When released in air of temperature \( 20^\circ\text{C} \) and density \( \rho_{\text{air}} = 1.20 \text{ kg/m}^3 \), it lifts a length \( h \) of string and then remains stationary as shown in Figure P14.33. We wish to find the length of string lifted by the balloon. (a) When the balloon remains stationary, what is the appropriate analysis model to describe it? (b) Write a force equation for the balloon from this model in terms of the buoyant force \( B \), the weight \( F_b \) of the balloon, the weight \( F_{\text{He}} \) of the helium, and the weight \( F_s \) of the segment of string of length \( h \). (c) Make an appropriate substitution for each of these forces and solve symbolically for the mass \( m_s \) of the segment of string of length \( h \) in terms of \( m_b \), \( r \), \( \rho_{\text{air}} \), and the density of helium \( \rho_{\text{He}} \). (d) Find the numerical value of the mass \( m_s \). (e) Find the length \( h \) numerically.