Studio Session 8

Buoyancy and the equation of continuity

Liquids are incompressible.  Their density ρ = mass/volume is constant.  When a liquid flows through a pipe, conservation of mass leads to the equation of continuity.
Consider the flow of a fluid through a pipe with varying cross sectional area A.
The volume V1 of liquid flowing into the pipe equals the volumeV2 flowing out of the pipe per unit time.
V1/Δt = V2/Δt,  A1Δx1/Δt = A2Δx2/Δt,  A1v1 = A2v2.


For the pipe we write the equation of continuity as A1v1 = A2v2, or Q = Av = constant.  Q is called the volume flow rate.

In this session, you will try to verify the equation of continuity for water flowing out of the bottom of an elevated can through a small-diameter hose.  You will also measure the fraction of the ordered energy that is lost because of friction.  In another experiment you will determine the density of a metal block using only a balance, by applying Archimedes' principle.

Equipment needed:

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will form the basis of your lab report.  Address the points highlighted in blue.  Answer all questions.

Experiment 1

Each table will work as a group on this experiment.

One end of a rubber hose is attached to a can with a spout in the bottom.  The can has an inside diameter of d1 = 9.8 cm and the rubber hose has an inside diameter of d2 = 0.8 cm.  The other end of the hose is taped to a horizontal rod a certain distance below the can.  While one member of your group plugs the hose with a finger, another member fills the can with water.  Then you will allow the water to drain from the can through the hose into a catch pan.  You will measure the speed with which the water level in the can drops, and the speed with which the water emerges from the hose.

image       image

Set up the apparatus on the floor, not on the table. 

Table 1

can diameter  (cm) 9.8
hose diameter (cm) 0.8
tape length (cm) 5
∆y (cm)  
h (cm)  
∆t (s)  
∆x (cm)  

Fill up the can with water to above the upper end of the black tape without allowing water to drain.  Distribute tasks to be performed after you start draining the water.

Drain the water while making measurements and record your measurements of ∆x and ∆t in the table.

Data Analysis:

Fill in table 2.

Table 2

v1 = 5/∆t (cm/s)  
v2 = ∆x/(2∆y/980)1/2 (cm/s)  
A1v1 = (πd12/4)v1  (cm3/s)  
A2v2 = (πd22/4)v2  (cm3/s)  
R = (v22 - v12)/(2*980*h)  


Experiment 2

Each group will perform their own experiment.
Each group has two metal blocks, a small metal can, a can with a spout, and a container to hold extra water.  Each group will determine the density of the metal blocks using a balance only.

Record all your measurements in a table in your log.

  wblock wempty small can wsmall can with water wdisplaced water wbuoyant Vwater ρblock
Block 1              
Block 2              

For each block:

imageSet up the balance.  Turn it on and zero it.  Determine the weight of the metal block.
Note:  The readout of the balance is in kg, the weight is m*g.

imageDetermine the weight of the small metal can.

imageFill the can with the spout with water until it overflows.  Catch the overflow in the small can and discard it.
Then lower the metal block into the water and catch the displaced water with the small can.
Determine the weight of the small metal can with the displaced water in it using the balance.
Calculate the weight of the displaced water by subtracting the weight of the can.
Now we will calculate the buoyant force and compare it to the weight of the displaced water.

imageTo make a measurement of the buoyant force, place the can with the spout onto the balance.
Make sure there is enough water in it so you can submerge the metal block, but not enough to cause it to overflow onto the balance when you do submerge it.
Zero the balance with the can on it.

imageGently lower the block into the water.  Make sure it is totally submerged but is not touching the walls or bottom of the can.  The water is pushing up on the block with a force equal to wbuoyant.  The scale now measures the reaction force with which the block is pushing down, which is equal in magnitude to the buoyant force by Newton's third law.  Record wbuoyant.

Given the mass of the displaced water calculate its volume. 
The density of water is ρ = m/V = 1000 kg/m3, so V = m/(1000 kg/m3).
The volume of the metal block is equal to the volume of the displaced water.
Calculate the density ρblock = mblock/Vwater.

Can you identify the materials your blocks are made of using your result and the table below and the appearance of the blocks?

Material Density (kg/m3)
Aluminum 2.7*103
Brass 8.32*103
Lead 11.3*103
Copper 8.96*103
Steel 7.65*103
Water 1.0*103


Convert your log into a session report, certify with you signature that you have actively participated, and hand it to your instructor.