Buoyant Force and Archimede's principle

[Ering]

Homework Statement

A balloon used for a physics experiment has a radius of 3.15m and is filled with helium. The total mass of the balloon is 15.2kg and the density of the surrounding air is 1.28 kg/m³.
a. How much buoyant force is acting on the balloon?
b. When the balloon is released from the ground, what is the acceleration of the balloon?
c. As the balloon rises, what happens to the radius of the balloon? Explain.

Homework Equations

F_B = W_Dis (buoyant force equals weight of the fluid displaced)
Archimede's principle = an immersed object is buoyed up by a force equal to the weight of the fluid it displaces

P = F/A (pressure equals force divided by area)

P₂ = P₁ + ρgh
P2 = pressure at top
P1 = pressure at bottom
ρ = density
g = gravity
h = depth/height

ρ = m/v (density equals mass divided by volume)

P₁ + 1/2 ρV₁² + ρgy₁ = P₂ + 1/2 ρV₂² + ρgy₂
V = speed
y = height/depth
P = pressure
ρ = density
g = gravity

The Attempt at a Solution

a. [/B]Buoyant force(Fb) = ?
I drew a free body diagram of the balloon, with Fb pointing up, and Wb, weight of the balloon, pointing down.
Summed forces:
ΣF_y = ma_y
Fb - Wb = 0 (since acceleration = 0)
Fb = Wb
Fb = Mb(g)
Fb = (15.2 kg)(9.81 m/s²)
Fb = 149.112 N

I'm not sure if this is correct or not? Are there any other forces acting on the balloon that I missed?

b. Acceleration of balloon (a_b) = ?

Fb - Wb = ma
a = (Fb - Wb) / m

That's as far as I've gotten with this one - I'm not sure if this is the correct way to go about finding the acceleration. If so, for the m in the equation, would i be using the mass of the balloon?

I feel like there's something I need to add since the density of the air was given in the problem, but I'm not sure where that fits in?

Any help would be great,
thanks!

 

[gneill]

Did your FBD include the tension in the rope holding the balloon down? It's only because it's initially tethered that it doesn't accelerate. So your equation:

Fb - Wb = 0 (since acceleration = 0)

is not correct.

Use Archimede's principle to calculate the buoyant force. That's where the density of the air comes into play.