Calculating Vol of Helium to Lift 400kg Payload to 8000m

[Paymemoney]

Homework Statement

How many cubic meters of helium are required to lift a balloon with 400kg payload to a height of 8000m where the density of air is 0.460kg/m^3 and the density of helium is 0.180kg/m^3. Assume that the balloon maintains a constant volume.

Homework Equations

$$\rho = M/V$$

$$F=PA$$

$$F=\rho*g*V$$

$$P=\rho * g * h$$

The Attempt at a Solution

Firstly i found the volume of air which is 869.59m^3 and then from this i found the Buoyancy force for air. The Buoyancy force for air is 3920.02156N

Now to find the Volume of helium

$$F=\rho*g*V$$

$$3920.02156 = 0.180 * 9.8 * V$$

$$V =2222.234m^3$$

However this is the wrong answer, someone tell me what is the problem with my solution?

P.S

 

[Filip Larsen]

I'm not sure how you can start out with a volume like that since that is what this problem directs you to calculate.

Try make an equation that describes the buoyancy force as a function of the so-far unknown volume (hint: this only needs to involve the two densities, the gravity constant, and the volume). Then think about what forces that are acting on the balloon (hint: there are two) and how they must relate to each other if the balloon should be able to just float at the given altitude. You should end up with an equation involving the volume as the only unknown for which you can then solve.

 

[Paymemoney]

That means i would this equation:

$$\rho_h * V_h = \rho_a * V_a$$

The two forces would be the Buoyancy force and mg.

 

[Filip Larsen]

There is only one volume, the volume of the balloon. The buoyancy force can be calculated as the weight of the air the balloon has displaced minus the weight of the helium, both calculated using the same volume (notice that mass and weight are not the same, weight is a force and is mass times the gravitational constant).

You are correct about the two forces involved.

 

[Paymemoney]

so if those are the two forces then:

$$F_n = F_b - mg$$
$$F_n = \rho_a * V * g - \rho_b * V * g$$
$$F_n = Vg(\rho_a - \rho_b)$$

If i sub in the values i get:

$$F_n = 9.8V(0.460 - 0.180)$$
$$F_n = 2.744V$$

Now how can i find the Volume?

 

[ideasrule]

You know that Fn has to be equal to the weight of the payload for the payload to be barely lifted.

 

[Paymemoney]

But isn't Net Force equal to the number of forces acting on the object?

 

[Filip Larsen]

You know the net force is the sum of the two forces F_g and F_b, where F_b is expressed using the yet unknown volume, but since the balloon is floating without accelerating up or down (i.e. it stays at altitude 8000m) you also know from Newtons 2nd law that the net force must be zero. This gives you an equation F_g + F_b = 0 which you can solve for the unknown volume. Take care to get the sign of the two forces correct, i.e. decide which direction (up or down) you consider positive and put signs on the two forces according to that.

 

[Paymemoney]

so is this what it will look like:

$$F_b - F_g = F_b - mg$$

 

[Filip Larsen]

Yes, now you just need to equate this to zero, insert the expression for F_b and solve for the unknown volume.

 

[Paymemoney]

$$\rho * g * V - mg = 2.744V$$

$$4.508V - 3920 = 2.744V$$

$$V= \frac{3920}{1.764}$$

$$V=2222.22m^3$$

This is incorrect, what have i done wrong?

 

[Filip Larsen]

You seem to keep using some of the old (incorrect) equations that involve V on both sides of the equation.

Start from scratch. You have two forces, the gravity force on the payload F_g = m*g and the opposite acting buoyancy force F_b = (rho_a-rho_b)*g*V. Now you insert this into the equation F_b-F_g = 0 and solve for V.