Question

A rock is found to weigh `w` in air when suspended from a spring scale. When completely submerged in water while attached to the spring scale, it weighs `w_("submerged")` as shown in Fig. Find the density of the rock `rho_("rock")` in terms of Rock the scale reading and the density of water

Answer 1

Let the volume of the rock be `V`, then magnitude of its weight.
`w=rhoi_("rock")Vg`
In water, rock experiences an additional buoyant force with a magnitude equal to that the weight of the water displaced by the system. Buoyant force has magnitude

`F_("buoy")=rho_("water")Vg`
The new scale reading is the total downward force on the system when it is the water
`w_("submerge")=w-F_("buoy")-rho_("rock")Vg-rho_("water")Vg`
From the ratio `w_("submerge")/w=(rho_("rock")Vg-rho_("water")Vg)/(rho_("rock")Vg)`
`=(rho_("rock")-rho_("water"))/rho_("rock")`
solving for the density of the system (i.e the rock )
`rho_("rock")=w/(w-w_("submerge"))rho_("water")`