A helium-filled balloon (whose envelope has a mass of...

A helium-filled balloon (whose envelope has a mass of $m_{b}=0.250 \mathrm{~kg}$ ) is tied to a uniform string of length $\ell=2.00 \mathrm{~m}$ and mass $m=0.0500 \mathrm{~kg} .$ The balloon is spherical with a radius of $r=0.400 \mathrm{~m}$. When released in air of temperature $20^{\circ} \mathrm{C}$ and density $\rho_{\text {air }}$ $=1.20 \mathrm{~kg} / \mathrm{m}^{3},$ it lifts a length $h$ of string and then remains stationary as shown in Figure $\mathrm{P} 14.33 .$ 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 $F_{\mathrm{He}}$ 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 $\rho_{\mathrm{He}^{-}}$ (d) Find the numerical value of the mass $m_{s}$. (e) Find the length $h$ numerically.

Key Concepts

Buoyant Force

This refers to the upward force exerted by a fluid on a body immersed in it, as described by Archimedes’ principle. It depends on the density of the fluid and the volume of fluid displaced by the object. In problems involving gases and balloons, calculating the buoyant force is essential to determine whether the object will float or sink.

Gravitational Force (Weight)

Weight is the force due to gravity acting on an object’s mass. In this context, it is important to consider the weight of all components of the system (the balloon envelope, the helium inside, and any attached string) as these forces act downwards and oppose the buoyant force.

Static Equilibrium

Static equilibrium occurs when the sum of forces acting on an object is zero, resulting in no net acceleration. In the scenario of the balloon lifting a length of string, the upward buoyant force and the downward gravitational forces balance each other, allowing the system to remain stationary.

Free-Body Diagram and Force Balance

A free-body diagram is a graphical representation used to visualize all the forces acting on a single object or system. For the balloon problem, writing a force balance equation based on the free-body diagram is crucial to relate the buoyant force and all the weight forces, which in turn permits setting up the equations required to solve for unknown quantities.

Density and Volume Relationships

Density, defined as mass per unit volume, plays a pivotal role in problems involving buoyancy and weight. It allows the conversion of volumes (such as that of the spherical balloon) into masses, thus enabling the calculation of the corresponding weight and buoyant force. These relationships are particularly useful when substituting known expressions into the force balance equations.

Transcript

00:01 Hi there, so for this problem, we're given the mass of the balloon and that mass is equal to 0 .25 kilograms.

00:14 We're also given the length of this string and that is equal to two meters.

00:21 And we are given the mass of this string, which is 0 .05 kilograms.

00:32 Now the balloon is spherical and the radius for that is equal to 0 .4 meters and the temperature and density.

00:43 So the temperature is 20 celsius degrees and the density for air is given and that is equal to 1 .2 kilograms per cubic meter.

00:56 It lifts a length age of a stream as you can see from the figure and remains stationary.

01:05 So with that said, for part a of this problem, we are asked about when the balloon remains stationary, what is the upper pit analysis model to describe it? and for that, we just, the model that we need to apply is a particle in equilibrium model.

01:40 So that's the solution for part a of this problem.

01:49 Now, for part b, we are asked about to write a force equation for the balloon from this model in terms of the buoyant force b, the weight f of f of f of the balloon, the weight of the balloon, and the weight f of x of the settlement of the string of length h.

02:15 So what we need to do is to write the expressions for each of the terms in the force equation.

02:21 So first of all, we know that the buoyant force is defined as the density of error, this times the volume, times the acceleration due to gravity.

02:32 Remember that this is the volume of a sphere.

02:35 So that will be the density of air times the volume of a sphere that is 4 divided by 3.

02:40 This times pi times the radius to the 3 times the acceleration due to the gravity.

02:47 And the force for the balloon is just its weight.

02:57 Now the force for ilion is going to be the density for alien.

03:04 This times the volume, this times the acceleration due to gravity.

03:08 So that will be the density for alien.

03:11 This 4 divided by 3 times pi times the radius to the 3 times the acceleration due to gravity.

03:23 Now, and the force f of s, which is the force in the string, is just the mass of the string times the acceleration due to gravity.

03:31 Now for this, we know that the mass of the stream is just the mass that we are given times the high h divided by the length l.

03:40 Therefore, from this, adding this to a force equation, we will have the density of earth times the volume, times the acceleration due to gravity, minus the mass of the balloon times the acceleration due to the gravity, minus the density for alien, this times the volume times the acceleration due to the gravity, minus the mass of the stream, times the acceleration to gravity, and then we set this equal to zero.

04:11 We can cancel all of these acceleration due to gravity.

04:16 And then solve for the mass of the strain.

04:20 We will have the following.

04:21 We'll have the density of the air minus the density of alien.

04:37 This times the volume that we know is 4 divided by 3 times pi times the radius to the 3...

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