A cylindrical bottle of oil-and-vinegar salad dressing whose volume is 1/3 vinegar (p=1.01 g/cm^3) and 2/3 oil (p=.910 g/cm^3) is at rest on a table. Initially the oil and the vinegar are separated, with the oil floating on top of the vinegar. The bottle is shaken so that the oil and vinegar mix uniformly, and the bottle is returned to the table. How has the height of the center of mass of the salad dressing changed as a result of the mixing?
a. it is higher
b. it is the same
c. not enough info
d. it is lower
Lower
To determine how the height of the center of mass of the salad dressing has changed as a result of the mixing, we need to consider the densities and relative volumes of the oil and vinegar.
1/3 of the volume of the salad dressing is vinegar, which has a density of 1.01 g/cm^3. Since the vinegar is denser than the oil, it will sink to the bottom when the bottle is at rest.
2/3 of the volume is oil, which has a density of 0.910 g/cm^3. When the bottle is shaken, the oil and vinegar mix uniformly, forming a emulsion.
When a heavier component (vinegar) sinks beneath a lighter component (oil) in a mixture, the height of the center of mass will be closer to the heavier component.
Therefore, as a result of the mixing, the height of the center of mass will be lower.
The correct answer is (d) it is lower.
To determine how the height of the center of mass of the salad dressing changes as a result of the mixing, let's analyze the scenario.
Before the mixing, we have a cylindrical bottle of oil-and-vinegar salad dressing. The volume ratio of vinegar to oil is 1:2, which means that if we divide the volume of the dressing into three equal parts, one part is vinegar and two parts are oil.
During the mixing, the oil and vinegar mix uniformly. This means that the vinegar and oil are now distributed evenly throughout the entire volume of the dressing.
The density of vinegar is 1.01 g/cm³, while the density of oil is 0.910 g/cm³. Since the oil is less dense than the vinegar, it remains on top of the vinegar even after the mixing. Therefore, the oil continues to occupy the top two-thirds of the dressing, while the vinegar occupies the bottom one-third.
When the oil and vinegar are separated, the center of mass of the dressing is closer to the top due to the higher concentration of oil on top.
After the mixing, even though the oil and vinegar are distributed uniformly, the center of mass will still be closer to the top of the dressing because the top two-thirds is occupied by oil, which has a lower density than the vinegar in the bottom one-third.
As a result, the height of the center of mass of the salad dressing remains the same or may have slightly increased due to the redistribution of the oil. Therefore, the correct answer is (b) it is the same.