Question

A solid cylinder of uniform density of 0.85 g/cm3 floats in a glass of water tinted light blue by food coloring.

Its circular surfaces are horizontal. What effect will the following changes, each made to the initial system, have on X, the height of the upper surface above the water? The liquids added do not mix with the water, and the cylinder never hits the bottom.

Answer choices are: I-Increase, D-Decrease, N-No Change

(a) The cylinder is replaced with one that has the same density and height, but 1.5× the diameter.

(b) The cylinder is replaced with one that has the same density and diameter, but with 1.5× the height.

(c) A liquid with a density of 1.06 g/cm3 is poured into the glass.

(d) A liquid with a density of 0.76 g/cm3 is poured into the glass.

(e) Some of the water is removed from the glass.

(f) The cylinder is replaced with one that has the same height and diameter, but with density of 0.83 g/cm3.

Answer 1

The buoyant force acting on the cylinder is, F pAhg Here, A is the cross-sectional area of the cylinder, h is the height of the cylinder, p is the density of the cylinder, and g is acceleration due to gravity C This buoyant force is also equal to the volume of the fluid displaced. - oh(A-x)g Here, o is the density of the fluid. C Equate the above two equations and solve for x. PAhg = oA(h-xg _ ph σh- σχ (σ-ρ)h So, the distance x depends on the density of the fluid, density of the cylinder and the height of the cylinder (a) The density of the cylinder is same and distance x is independent of the diameter of the cylinder. Therefore, there will be no change in the distance x. Hence, the correct answer choice is N-No change. (b) Now the height is changing keeping the density same. As the distance x is directly proportional to the height, the distance x will increase Hence, the correct answer choice is I-Increase (c) The density of the added liquid is greater than that of water and it does not mix with the water. So, the liquid will settle down and there will be no change in the distance x Hence, the correct answer choice is N-No change

(d) The density of the added liquid is less than that of water and it does not mix with the water So, the liquid will not settle down and the distance x will change The change in distance x can be determined as follows: pAhg o'Axg+ oA(h-x)g ph a'xoh-ax Here, o is the density of the added liquid. From the above relation it is clear, that on adding the liquid of the density less than that of water, the denominator term become small and so the value ofx will increase. Hence, the correct answer choice is I-Increase. (e) On removing some of the water inside the glass, the height of the water column will decrease, but the value of x does not depend on the height of the water column. So, there will be no change in the distance x Hence, the correct answer choice is N-No change (f) The density of the new cylinder is smaller than that of the earlier one. So, the numerator term will increase. Therefore, the value ofx will increase Hence, the correct answer choice is I-Increase