Question

A small solid sphere of mass M₀, of radius R₀, and of uniform density ρ₀ is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.
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R F U R or U F or U R or F or U  The new sphere has density ρ = ρ₀ and radius R > R₀
R F U R or U F or U R or F or U  The new sphere has radius R = R₀ and density ρ > ρ₀
R F U R or U F or U R or F or U  The new sphere has density ρ < ρ₀ and mass M = M₀
R F U R or U F or U R or F or U  The new sphere has radius R > R₀ and density ρ < ρ₀
R F U R or U F or U R or F or U  The new sphere has mass M = M₀ and radius R < R₀
R F U R or U F or U R or F or U  The new sphere has mass M < M₀ and density ρ = ρ₀

Answer 1

To determine the effect on the water level L we use archimedes principle "the volume of the immersed object will be exactly equal to the volume of the displaced fluid"

(i) The new sphere has density ρ = ρ₀ and radius R > R₀

as new radius is more , so the volume is increased . This will more water to be displaced hence water level L will rise / increase

(ii)The new sphere has radius R = R₀ and density ρ > ρ₀

Using the principle " Any object, totally or partially immersed in a fluid or liquid, is buoyed up by a force equal to the weight of the fluid displaced by the object."

The new sphere has radius R > R₀ and density ρ < ρ₀

As the density is increased,weight is also increased for same radius, so with respect to initial condition there is comparatively more downward(weight ) force, hence  water level L will Rise / increase

(iii)  The new sphere has density ρ < ρ₀ and mass M = M₀

this indirectly means volume is increased . This will more water to be displaced hence water level L will Rise / increase

(iv) The new sphere has radius R > R₀ and density ρ < ρ₀

as R is more , more buoyant will act and as density is reduced less downward(weight ) force. BOTH these factor will cause  water level L to FALL

(V) The new sphere has mass M = M₀ and radius R < R_0,

as volume is reduced less  buoyant will act hence water level L will FALL

(VI) The new sphere has mass M < M₀ and density ρ = ρ₀

asdownward(weight ) force is reduced hence water level L to RISE