A light spring of constant k = 95.0 N/m is attached vertically to a table (figure (a)). A 2.70-g balloon is filled with helium (density = 0.179 kg/m3) to a volume of 4.00 m3 and is then connected to the spring, causing the spring to stretch as shown in figure (b). Determine the extension distance L when the balloon is in equilibrium. (The density of air is 1.29 kg/m3.)


m

bouyant force=weight of air displaced-weight of balloon.

= denAir*4m^3-densityH3*4m^3 -.027(9.8)
= 4(1.29kg-.179kg) - .027g=4*1.11 N-.027*9.8N

stretch=force/k=bouyantforceabove/95N/m

To determine the extension distance L when the balloon is in equilibrium, we need to consider the forces acting on the system.

1. Gravitational force on the balloon:
The weight of the balloon is given by the formula:
Weight = density * volume * acceleration due to gravity

Given:
Density of helium (ρ_helium) = 0.179 kg/m^3
Volume of the balloon (V_balloon) = 4.00 m^3
Acceleration due to gravity (g) = 9.8 m/s^2

Weight of the balloon = ρ_helium * V_balloon * g

2. Buoyant force on the balloon:
The buoyant force on the balloon is given by the formula:
Buoyant force = density of fluid * volume submerged * acceleration due to gravity

Given:
Density of air (ρ_air) = 1.29 kg/m^3

Buoyant force on the balloon = ρ_air * V_balloon * g

3. Force exerted by the spring:
The force exerted by the spring is given by Hooke's law:
Force = spring constant * extension distance

Given:
Spring constant (k) = 95.0 N/m

Force exerted by the spring = k * L

Now, for the balloon to be in equilibrium, the forces exerted by the spring, gravity, and buoyancy must balance each other.

So, we have the equation:
Force exerted by the spring = Weight of the balloon + Buoyant force on the balloon

k * L = ρ_helium * V_balloon * g + ρ_air * V_balloon * g

Substituting the given values into the equation, we can solve for the extension distance L.

To determine the extension distance L when the balloon is in equilibrium, we need to consider the forces acting on the system.

Let's break the problem down step by step:

Step 1: Calculate the weight of the balloon:
The weight of an object can be calculated using the formula:
weight = mass * gravity

The mass of the balloon, m, is given as 2.70 g. We need to convert it to kilograms:
mass = 2.70 g = 2.70 * 10^-3 kg

The acceleration due to gravity, g, is a constant value of 9.8 m/s^2.

Now we can calculate the weight of the balloon:
weight = mass * gravity = 2.70 * 10^-3 kg * 9.8 m/s^2

Step 2: Calculate the buoyant force on the balloon:
The buoyant force on an object submerged in a fluid can be calculated using the formula:
buoyant force = density of fluid * volume of fluid displaced * gravity

The density of air, ρ_air, is given as 1.29 kg/m^3.
The volume of the balloon, V, is given as 4.00 m^3.

Now we can calculate the buoyant force on the balloon:
buoyant force = density of air * volume of balloon * gravity = 1.29 kg/m^3 * 4.00 m^3 * 9.8 m/s^2

Step 3: Calculate the net force on the balloon:
The net force on the balloon is the difference between the weight and the buoyant force:
net force = weight of balloon - buoyant force

Step 4: Calculate the spring force:
The spring force, Fs, can be calculated using Hooke's Law formula:
Fs = k * L

The spring constant, k, is given as 95.0 N/m.
The extension distance, L, is what we are trying to find.

Step 5: Set up the equation for equilibrium:
In equilibrium, the net force on the balloon must be zero. So we can set up the equation:
net force = spring force

weight of balloon - buoyant force = k * L

Step 6: Solve for L:
Now we can substitute the calculated values into the equation and solve for L:
(mass * gravity) - (density of air * volume of balloon * gravity) = k * L

Simplify the equation:
(mass * gravity) - (density of air * volume of balloon * gravity) = 95.0 N/m * L

Finally, solve for L:
L = [(mass * gravity) - (density of air * volume of balloon * gravity)] / 95.0 N/m

Substitute the given values into the equation to find the value of L.