One day when you come into physics lab you find several plastic hemispheres floa

Question

One day when you come into physics lab you find several plastic hemispheres floating like boats in a tank of fresh water. Each lab group is challenged to determine the heaviest rock that can be placed in the bottom of a plastic boat without sinking it. You get one try. Sinking the boat gets you no points, and the maximum number of points goes to the group that can place the heaviest rock without sinking. You begin by measuring one of the hemispheres, finding that it has a mass of 20g and a diameter of 9.0cm .

What is the mass of the heaviest rock that, in perfectly still water, won't sink the plastic boat?

Explanation / Answer

e bouyant force of the water equals the weight of the water displaced by the hemisphere. The max. force will occur when the hemisphere sinks until it is level with the water surface, as this will displace the most amount of water.

The volume of a hemisphere is (pi/12)D^3

Water has a density of 1000kg per cubic meter so the mass of water displaced by the hemisphere is M= (density)x(volume) = (1000)(pi/12)D^3

And the weight of this mass of water is F= g(1000)(pi/12)D^3

This is the upward bouyant force of the water and you set it equal to the weight of the boat plus the unknown weight of the rock;

Mg + mg = g(1000)(pi/12)D^3

THe mass of the rock is then;

M = -m + (1000)(pi/12)D^3

= - .020 + (1000)(pi/12)(.09)^3

= .170 kg

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