Help on submerged ball problem given tension and mass?

Question

A 3.5 g hollow plastic ball is attached by a thread to the bottom of a beaker. When the beaker is filled with water so that the ball is totally submerged, the tension in the thread is 3.0 multiplied by 10-2 N. Determine the diameter of the ball.

______ cm

Answer 1

The tension on the string is the weight of the displaced water minus the weight of the plastic ball mg.  The weight of the displaced water will be the water's mass times g, the mass being the volume of the plastic ball V times the density of water ρ, which we will take as 1000 kg/m³.  We will also take g to be 9.8 m/s².

In other words,

T = ρVg − mg

so V = (T/g + m) / ρ = 4/3πr³ = 1/6πd³

so d = [6/π (T/g + m) / ρ]^(1/3)

= [6/π (3.0e-2 N / 9.8 m/s² + 3.5e-3 kg) / 1000 kg/m³]^(1/3)

= 0.0232 m = 2.32 cm