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 23 g and a diameter of 8.4 cm . What is the mass of the heaviest rock that, in perfectly still water, won't sink the plastic boat?

Answer 1

To solve this problem we will first proceed to find the volume of the hemisphere, from there we will obtain the mass of the density through the relation of density. Finally the mass of the stone will be given between the difference in the mass given in the statement and the one found, that is

The volume of a Sphere is

$V = \frac{4}{3} \pi r^3$

Then the volume of a hemisphere is

$V =\frac{1}{2} \frac{4}{3} \pi r^3$

With the values we have that the Volume is

$V =\frac{1}{2} \frac{4}{3} \pi (8.4/2)^3$

$V = 155.17cm^3$

Density of water is

$\rho = 1g/cm^3$

And we know that

$\text{Mass of water displaced} = \text{Density of water}\times \text{Volume of hemisphere}$

$m = 1g/cm^3 * 155.17cm^3$

$m = 155.17g$

So the net mass is

$\Delta m = m_s-m_w$

$\Delta m = 155.17-23$

$\Delta m = 132.17g$

Therefore the mass of heaviest rock is 132.17g or 0.132kg