Question

A small solid sphere of mass M₀, of radius R₀, and of uniform density ?₀ 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.
Read it to me

R F U R or U F or U R or F or U  The new sphere has radius R = R₀ and density ? < ?₀
R F U R or U F or U R or F or U  The new sphere has density ? = ?₀ and mass M < M₀
R F U R or U F or U R or F or U  The new sphere has density ? > ?₀ 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 ? < ?₀
R F U R or U F or U R or F or U  The new sphere has mass M > M₀ and density ? = ?₀
R F U R or U F or U R or F or U  The new sphere has density ? < ?₀ and mass M = M₀

Answer 1

A body that folats on water displaces only the amount of water of same mass

By assuming that all objects float,then the water will rise when the new object has higher mass.

Part-1

Rises-R

Same density means the same proportion of the sphere is submerged; since we have a bigger sphere, more water is displaced.

Part-2

Falls-F

(Reverse argument for 1)

Part-3

Unchanged-U

Same mass and larger radius mean two things: the density is lower, meaning it continues to float; and therefore the same amount of water is displaced. The float rides higher.

Part-4

Unchanged-U

Mass is unchanged = level unchanged

Assuming the ball still floats. If it becomes so dense that it sinks, then I don't think we have enough information to decide between U and F.

Part-5

Mass is unchanged = level unchanged

Part-6

Size decreased = mass decreased = level falls

R constant and density decreasing means mass is decreasing; object still floats, but less water is displaced.