A small solid sphere of mass M0, of radius R0, and of uniform density 0 is place

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. (For each statement below determine if it will Rise, Fall or remain unchanged)

1.The new sphere has mass M = M0 and density < 0

2. The new sphere has mass M = M0 and density > 0

3.The new sphere has density = 0 and mass M < M0

4.The new sphere has radius R = R0 and mass M > M0

5. The new sphere has density < 0 and radius R > R0

6.The new sphere has radius R > R0 and density = 0

Explanation / Answer

The water level in the bowl is dependent on the volume of water displaced. The more water that is displaced, the higher L. The volume of the liquid displaced is a function of the density of the liquid, and the weight (Fg = mg) of the object pushing into the water.
Volume = mass / density, and mass is weight, Fg, divided by acceleration due to gravity so
V = Fg / (g x d) so now we see that F is directly proportional to mass and indirectly proportional to density.

If M decreases the V decreases and F is correct
If density increases V decreases but an decrease in R causes a further decrease in V so F
Density unchanged, R increases increasing V so R is correct.
M unchanged, R decreases causing V to decrease so F
M unchanged, R increases so V increases causing R
R decreases causing V to decrease so F is correct.

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