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.

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

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

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

The new sphere has radius R = R₀ and mass M < M₀

The new sphere has mass M = M₀ and radius R < R₀

Answer 1

Change in the level of water in the bowl is decided by change in the volume of water displaced by the ball.

In case of a ball that is floating, i.e. it is only partially submerged in the water, the weight of the ball is exactly balanced by the upward buoyancy force exerted on the ball by the water displaced by the portion of the ball that is in water.

Now, recall that Archimedes principle says that the upward buoyancy force exerted on an object, when partially or fully immersed in a fluid, is equal to the weight of the fluid displaced by the object.

So, for a ball to float on the surface of water it has to displace just the amount of water whose weight is equal to the weight of the ball. Note that the ball in above question, having uniform mass density, can float only if its density is less than the density of water. If ball's density is larger than that of water, it will sink.

Now let us consider the situations mentioned in the question.

(1) For the same mass as M0, the mass of water displaced by the ball is same as before. Therefore no change in water level. -U

(2)The level rises because the weight of the ball is larger and needs to displace more water to get a buoyancy force that will

cancell its weight. -R

(3) Water level will depend on what is mass of the ball in the new situation. We need to know by how much the density is lowered when its radius is increased. So, need more information.

(4) F

(5) F

(6) Depends on the new radius.