Calculations and Analysis
Analysis questions from lab: Part 1
Which side of the pan balance is lifted as you poured in the water? Was the solid cylinder forced up by the buoyant force of the water?
The left side; yes, slightly.
What happened to the left pan as you added water to the hollow cylinder? What can you say about the volume of the water you added to the hollow cylinder and the volume of the water (in the beaker) displaced by the solid cylinder? According to Archimedes' principle, should these volumes be the same?
As water was added to the hollow cylinder, the left side of the pan was forced down. The volume of water added to the hollow cylinder to obtain balance was equal to the volume of water displaced by the solid cylinder; this is in accordance with Archimedes' principle.
The buoyant force on the hollow cylinder exerted by the water is proportional to the mass of the water displaced by the cylinder. The buoyant force is given by the equation
B = Δmg
Where Δm is the mass of the water displaced and g is the acceleration of gravity. This equation reveals the buoyant force to be 0.21 N, or 21,000 dynes.
We can find out if the cylinder sinks in the glycerin by calculating the sum of the forces on the cylinder. If the net force is positive, there is a downward force on the cylinder causing it to sink. If the net force is negative, there is an upward force on the cylinder causing it to float. We use the equation
Fnet = gVal(ρal – ρg)
Since we know the accepted value of the density of aluminum is 2.7 g/mL and the density of glycerin is 1.26 g/mL, the difference in densities is a positive value and there is a net downward force on the cylinder, causing it to sink.
Archimedes' principle states that the upward, buoyant force on a submerged or floating object is equal to the weight of the fluid displaced by that object. Expressed in terms of the density and volume of the fluid, the weight of the fluid is
B = ρVobjg
where Vobj is used because the volume of the fluid displaced is equal to the volume of the immersed object.
This equation can be used in conjunction with Newton's second law to calculate the net downward force on the object.
Fnet = mg – Wfluid
= gVobjρobj - gVobj ρfluid
= gVobj(ρobj -ρfluid)
There were some sources of error in the laboratory experiment. In part 3, the density of aluminum was determined by suspending a cylinder of aluminum from the edge of a 2 pan balance scale and then submersing it in water. The cylinder was counterbalanced by an equal standard mass on the opposite pan. The density of the cylinder was found from the volume of the displaced fluid and the amount of mass that had to be removed to obtain balance once the cylinder was submerged. In this part of the experiment, the assumption had to be made that the string from which the cylinder was suspended was massless. Of course, in reality the string did have some mass, and this introduces a source of error. In addition, human error may have factored into this part of the experiment if the scale was not properly calibrated.
The experiment confirmed Archimedes' principle to a high degree of accuracy. In part 1 of the experiment, Archimedes' principle was verified with a simple demonstration. A solid aluminum cylinder was suspended from the left pan of a 2 pan balance scale, and a hollow cylinder of equal volume was placed on the same pan as the solid cylinder was suspended from. The solid cylinder was then submersed in water, causing the left pan to rise; to balance it, water was also poured in the hollow cylinder. The amount of water required to balance the scale was found to be equal to the volume of water displaced by the solid cylinder, in accordance with Archimedes' principle.
In part 2, Archimedes' principle was used to determine the density of water, which was found to be 0.989 g/mL. This value differs from the accepted value of 0.998 g/mL by only 0.85%. In part 3, we used Archimedes' principle to determine the density of aluminum, which is accepted to be 2.7 g/mL. Our determination of 2.667 differs from the accepted value by only 1.21%. Both parts of the experiment yielded results that were well within the acceptable range.