A large balloon of mass 237 kg is filled with helium gas until its volume is 318 m3. Assume the density of air is 1.29 kg/m3 and the density of helium is 0.179 kg/m3.
(b) Calculate the buoyant force acting on the balloon.
________N
(c) Find the net force on the balloon and determine whether the balloon will rise or fall after it is released.
________N
(d) What maximum additional mass can the balloon support in equilibrium?
________kg
(e) What happens to the balloon if the mass of the load is less than the value calculated in part (d)?
(X)The balloon and its load will accelerate downward.
(X)The balloon and its load will remain stationary.
(X)The balloon and its load will accelerate upward.
(f) What limits the height to which the balloon can rise?
The buoyant force acting on the balloon is (p1V1 - p2V2)g
where V1 = V2 = 318 m3
p1 is the density of air
p2 is the density of helium
c) The net force is the buoyant force - m*g, where m is the mass of the balloon. If the net force is positive, the balloon will rise
d BF - m*g - m(additional)*g = 0
where BF is buoyant force, m is mass of the balloon, m(additional) is additional mass
(e) for any mass less than this, the balloon will have enough buoyant force to accelerate upward.
f) I think the density of air decreases with increasing height.
To calculate the buoyant force acting on the balloon, we need to determine the weight of the displaced air.
First, find the volume of air displaced by the balloon:
Volume of balloon = 318 m^3
Since the density of air is 1.29 kg/m^3, the mass of the displaced air is:
Mass of displaced air = density * volume
= 1.29 kg/m^3 * 318 m^3
= 408.42 kg
The buoyant force acting on the balloon is equal to the weight of the displaced air:
Buoyant force = mass of displaced air * gravitational acceleration
= 408.42 kg * 9.8 m/s^2
≈ 4002.876 N
So, the buoyant force acting on the balloon is approximately 4002.876 N.
To find the net force, we need to consider the weight of the balloon and the buoyant force.
The weight of the balloon is:
Weight of balloon = mass of balloon * gravitational acceleration
= 237 kg * 9.8 m/s^2
= 2322.6 N
The net force on the balloon is given by:
Net force = buoyant force - weight of balloon
= 4002.876 N - 2322.6 N
≈ 1680.276 N
If the net force is positive (greater than zero), the balloon will rise. If the net force is negative (less than zero), the balloon will fall. Therefore, since the net force is approximately 1680.276 N (greater than zero), the balloon will rise after it is released.
To find the maximum additional mass the balloon can support in equilibrium, we need to consider the buoyant force and the weight of the balloon and the additional mass.
Let the maximum additional mass be 'm':
Buoyant force = weight of balloon + weight of additional mass
4002.876 N = 2322.6 N + m * 9.8 m/s^2
Solving for 'm', we have:
m = (4002.876 N - 2322.6 N) / (9.8 m/s^2)
m ≈ 176.198 kg
Therefore, the maximum additional mass the balloon can support in equilibrium is approximately 176.198 kg.
If the mass of the load is less than the value calculated in part (d), the balloon and its load will accelerate upward. However, if the mass of the load exceeds this value, the net force will become negative, and the balloon will no longer be able to support the load. The balloon and its load will then accelerate downward.
The height to which the balloon can rise is limited by various factors such as air density, wind conditions, and the weight of the balloon itself. The higher the balloon rises, the lower the air density becomes. Eventually, the buoyant force may become equal to the weight of the balloon, resulting in the balloon reaching a maximum height where the net force becomes zero, and it remains at that altitude.
To solve this problem, we need to understand the concept of buoyant force and net force acting on the balloon.
(b) The buoyant force acting on the balloon can be calculated using the formula:
Buoyant Force = Weight of the fluid displaced
= density of fluid * volume of fluid displaced * gravitational acceleration
In this case, the fluid is air, so the density of air is 1.29 kg/m3. The volume of air displaced is equal to the volume of the balloon (318 m3). The gravitational acceleration is 9.8 m/s2.
Buoyant Force = 1.29 kg/m3 * 318 m3 * 9.8 m/s2
Calculating this will give you the buoyant force acting on the balloon in Newtons.
(c) The net force on the balloon can be calculated by subtracting the weight of the balloon from the buoyant force.
Weight of the balloon = mass of the balloon * gravitational acceleration
= 237 kg * 9.8 m/s2
Net Force = Buoyant Force - Weight of the balloon
If the net force is positive, the balloon will rise. If the net force is negative, the balloon will fall.
(d) The maximum additional mass the balloon can support in equilibrium is equal to the difference between the buoyant force and the weight of the balloon.
Maximum additional mass = (Buoyant Force - Weight of the balloon) / gravitational acceleration
This will give you the maximum additional mass that the balloon can support in kilograms.
(e) If the mass of the load is less than the maximum additional mass calculated in part (d), the balloon and its load will accelerate upward. So, the correct answer is (X) The balloon and its load will accelerate upward.
(f) The height to which the balloon can rise is limited by various factors, including the strength of the balloon material, the pressure and temperature changes as it rises, and atmospheric conditions. As the balloon rises, the air density decreases, and eventually, it reaches a point where the buoyant force is equal to the weight of the balloon and its load. At this point, the balloon reaches its maximum height and will not rise any further.