A 1.0-g balloon is filled with helium gas.When a mass of 4.0 g is attached to the balloon,it is seen to be static equilibrium in mid air.Assuming the balloon is spherical,what is its diameter?
A)20 cm
B)15 cm
C)25 cm
D)10 cm
E)40 cm

Question

Quizplus · Accepted Answer

The correct answer is A.The buoyant force on the balloon is equal to the weight of the air displaced by the balloon. The weight of the air displaced is equal to the mass of the air displaced times the acceleration due to gravity. The mass of the air displaced is equal to the volume of the balloon times the density of the air. The volume of the balloon is equal to 4/3πr^3, where r is the radius of the balloon. The density of the air is 1.29 kg/m^3. The acceleration due to gravity is 9.8 m/s^2.The buoyant force on the balloon is equal to:$$F_b = (4/3πr^3)(1.29 kg/m^3)(9.8 m/s^2)$$The weight of the balloon and the mass attached to it is equal to:$$F_w = (1.0 g + 4.0 g)(9.8 m/s^2) = 49 N$$In equilibrium, the buoyant force is equal to the weight of the balloon and the mass attached to it:$$F_b = F_w$$Substituting the expressions for the buoyant force and the weight into this equation, we get:$$(4/3πr^3)(1.29 kg/m^3)(9.8 m/s^2) = 49 N$$Solving for r, we get:$$r = 0.1 m = 10 cm$$Therefore, the diameter of the balloon is 20 cm.

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