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NATURE.M
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Homework Statement
Two balloons A and B are filled with He gas. They have identical radii of 10 cm. The balloons are made with rubber so that they can be charged by rubbing against hair. They are held together with insulating strings, whose mass and charge can be ignored for this problem. You can also ignore the mass of the balloons.
a. The magnitude of the charge on Balloon A is 80 nC and that on Balloon B is 40 nC. The strings attached to the balloons make angles θ1 and θ2 with the vertical. Which angle is larger?
b. Density of He gas is 0.17 kg/m3 and density of air is 1.20 kg/m3. Calculate the angular separation θ1 + θ2 at equilibrium. Hint: assume the angles are small. Let L, the length of the string be 1.0m.
The Attempt at a Solution
So basically, for part a. I found that if you draw a FBD of all the forces acting on the two balloons, all the forces are the same. (Vertical - Buoyant force, gravity, component of tension
Horizontal - electric force, component of tension). Therefore, the angles should be the same,
θ[itex]_{1}[/itex] = θ[itex]_{2}[/itex].
Then for part b, I take the equations for the forces acting on a balloon in the x direction and y direction (with the above mentioned forces). Then substituting one equation into another,
I get a pretty messy equation. But since we assume the angles are small, I let
sinθ[itex]_{1}[/itex] ≈ θ[itex]_{1}[/itex] and tanθ[itex]_{1}[/itex] ≈ θ[itex]_{1}[/itex].
Now assuming part b is correct, we know θ[itex]_{1}[/itex] + θ[itex]_{2}[/itex] = 2θ.
Then we simply just rearrange the equations and solve for θ. I obtain a fairly reasonable value of
≈3.2 deg. for theta.
Although everything seems to make sense to me, I'm not entirely confident with this answer.
If I made a mistake along the way, if you could please let me know its much appreciated.