Visualizing hydrostatic pressure distributions using piles of books

I typically try to avoid all those derived so-called simplifications for the force and line of action of a force on a submerged flat surface. Instead I teach the students that the force is the integral of the pressure over the area (with the pressure varying with depth) and that the line of action of the force can be calculated by taking moments about some point on the surface. This is how many text books introduce the topic but they then revert to the force being the pressure at the center of area times the area and use that horrible equation for the line of action that uses the second moment of area. I really dislike this equation as the geometry of applying the equation, particularly for a fully submerged angled surface, is very complex. I prefer to teach my students how to do the integrals. However, they also find this challenging so I use a simple visualization demonstration using piles of textbooks.

Equipment

All you need is a large number of mechanics textbooks (approximately 30), a yard stick, and a large flat surface such as a desk (I use the trolley that has the OHP on it in our class rooms). The topic of the books is obviously irrelevant. I am also typically not a big fan of textbooks. I find that the students do not use them the way they were written to be used so they can cause as many problems as they solve. I therefore use this demonstration to joke about having found a good use for all the textbooks in my office.

Demonstration

I start off by reviewing distributed loads from statics. I make 4 piles of books all of the same height with all the piles touching each other. This is a simple uniformly distributed load with the total load being equal to the weight of all the books.

Photo Dec 30, 6 06 10 PM

I then adjust the book piles so that the first pile has one book, the second has two books, the third has three and the forth pile has four. I then lay the yard stick across the top of the piles so that the stick represents a distributed load that increases linearly with distance along the table (or along the floor as in the pictures in this post).

Photo Dec 30, 6 07 42 PM

I then say that if you wanted to approximate the total load you would add up the load due to each pile.

F=Σ pile height x (weight/unit height of the books)

The approximation would improve if we used more piles of narrower books with the actual load being given by the sum of an infinite number of infinitesimally thin book piles. You can write this on the board as

F=limit (pile width →0) Σ pile height x (weight/unit height of the books ) (=∫fdx)

where f is the distributed load function.

To calculate the distance from some point (denoted by a) to the point of action of the force we calculate the moment about a (Ma) and divide that by the total load F. The moment can be calculated by adding up the moments due to the individual piles of books. Again the estimate is improved by using a large number of thin piles. For a load that increases linearly from zero, the force acts 2/3rds of the way along the distributed load. This often leads to confusion as this only applies to hydrostatic forces on a submerged surface if the surface is rectangular, and the top of the surface is at the free surface.

To illustrate this I ask the class to consider the hydrostatic pressure force on a non-rectangular flat surface. Without loss of generality, I consider a triangular surface that increases in width with depth. In this case I start with one book, next to that I build two piles of two books, then three piles of three books, and four piles of four books. The layout is shown in the images below. Again, I lay the yard stick over the pile showing the linear increase in height with distance from the first pile.

Photo Dec 30, 6 10 40 PM Photo Dec 30, 6 10 45 PM

This time, because each pile of books has a different width, the weight of the pile is related to the height of the pile and the transverse width of the pile, that is, the width of the surface. The force is given by

F=Σ pile height x (weight/unit height of the books) x # piles across

This is analogous to the force on the surface being given by

F=∫P da=∫ P(L) W(L) dL

where P(L) is the pressure as a function of distance, L, along the surface and W(L) is the width of the surface at a distance L along the surface.

The point of action can be calculated in the same manner as for the earlier distributed load. Simply take moments about the top of the surface (L=0) by summing the moment due to each pile of books

ML=0= Σ L x pile height x weight/unit height of the books x # piles across.

This is directly analogous to the integral form for the moment given by.

ML=0=∫ L P(L) W(L) dL.

The distance to the point of action is D= ML=0/F. In general this is not 2/3rds of the way along the distributed load.

Note

You can also use this demonstration to discuss numerical approximation of integrals. The nature of the integral does not matter; simply draw the analogy between the finite thickness slices (book piles) and the limit of infinitesimally thin book piles (integration).

Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Thanks to my father-in-law for lending me his books and floor for the photos.

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Video for drag, dimensional analysis, and coffee filters

Here is a short video showing a single coffee filter dropped from 3.5 feet and two stacked filters dropped from 5 feet. It goes with version one of the ‘drag, dimensional analysis, and coffee filters‘ demonstration.

It took about 10 tries to get it this close. (full video)

coffee

You can also use paper cup cake holders if you are not a coffee drinker. This one only took two attempts, though my helpers had been practicing with the coffee filters (full video)

CupCake

You may need to click on the GIF to get it to animate.

Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu).

Thanks to my two sisters-in-law, Helen and Leila, for helping with the video.

Drag, dimensional analysis, and coffee filters

This is another demonstration that Ben Sill showed me. It is very simple and can be used to demonstrate drag, dimensional analysis, experimental measurement, and data presentation.

Equipment

All you need is a tape measure, a stop watch, a packet of coffee filters and a couple of student volunteers.

coffee

Demonstration

Version 1:

  1. Give one student a single coffee filter and another student two filters stacked together.
  2. Have the students drop the filters at the same time from 5 feet above the ground. The 2 stacked filters will land first.
  3. Have the student with the single filter move their release point down a little bit and repeat the simultaneous drop.
  4. Keep moving the single filter release point down until the single filter and double filter land at the same time, and measure the release height (it should be about 3.5 feet).

Version 2:

  1. Have a student drop a single filter from 5 feet and a second student measure the time taken for it to fall to the floor.
  2. Add a second filter to the stack, repeat the drop, and measure the time to reach the floor.
  3. Keep adding more filters to the stack and measuring the drop time until you have about 4-5 different drop times (the drop times get so short after about 4 filters that they are too hard to measure).
  4. Plot drop time versus number of filters in the stack.

Class discussion

This is a great demonstration (particularly version 2) to use as a basis for discussing measurement errors, and the need to take multiple measurements. There are a lot of sources of measurement error including:

  1. Differences in release time in version 1
  2. Error in starting and stopping the stopwatch at release and at impact respectively.
  3. Deformation of the coffee filter between drops

If you have time you can re-do all the drops a few more times and take the average of the times. However, once you have more than about 4 filters the drop time is small enough that the reaction time error is large compared to the difference in drop time you get from adding an additional filter.

Analysis

There are two basic assumptions in the analysis.

  1. The filter accelerates to its terminal velocity very quickly. Therefore, the terminal velocity is approximately the distance fallen divided by the time taken.
  2. The drag coefficient, Cd, is constant.

Drag analysis:

If you have covered bluff body drag then you can do the following.

When at its terminal velocity, the filter’s weight is balanced by the drag force. Therefore, you can write

mg=ρU2CdA/2.                  (1)

The time taken for the filters to drop a distance h is

T=h/U=h(ρCdA/2mg)1/2.

The only things that vary are the mass and the drop height (the area is constant as the filters stack very closely together). Therefore, the drop time is given by

T=ch/m1/2                          (2)

where c=(ρCdA/2g)1/2 is an unknown constant.

For version 1 of the demonstration the time taken for the single filter and the double filter to hit the floor is identical and given by:

T=c(5/(2m)1/2)=ch1/m1/2

where h1 is the release height of the single filter and the 5 is the 5 feet drop height for the double filter. Therefore, the height that the single filter must be dropped from to land at the same time as the double filter dropped from 5 feet is given by

h=5(m/2m)1/2=3.5 feet.

For version 2 of the demonstration h is constant so T~m-1/2. Plot the time versus number of filters on both a linear-linear scale and a log-log scale. The linear scale plot should show a non-linear decreasing function. This log-log scale plot should be a line of slope -1/2. Example plots for the linear and log scales are shown below. The use of the log-log scale plot illustrates an alternate method of data presentation. It shows the power-law nature of the relationship between time and mass. You can use this to lead into a discussion with the class  on how to identify power law relationships (log-log scale plots) and exponential (semi-log scale plots) in experimental data.

lin-lin-coffee

linear-linear scale plot of time taken to drop 5 feet as a function of the number of nested coffee filters.

log-log-coffee

log-log scale plot of time taken to drop 5 feet as a function of the number of nested coffee filters. Also shown is a line of slope -1/2 (see equation (2) above).

Dimensional analysis:

If you have not yet covered bluff body drag but have covered dimensional analysis then you can derive the drag force scaling as an application of dimensional analysis. Assuming that the drag is controlled by the flow separation around the filter, then viscosity plays no significant role. In this case, the drag force is a function of the object area, density of air, and the velocity of the object. that is:

D=D(A,ρ,U).

The dimensions of these parameters are

[D]=MLT-2, [A]=L2, [ρ]=ML-3, and [U]=LT-1

There are 4 parameters and 3 independent dimensions so we can only form 1 non-dimensional group. Standard dimensional analysis calculations lead to

Π1=D/ρU2A.

From here you can write the balance of weight (mg) and drag as given in equation (1) and follow the rest of the analysis outlined above. The only thing different from the previous analysis is that the 2 and the Cd are missing from the equations and are replaced by another unknown constant.

don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu).

Videos of “Visualizing streak lines, path lines, and streamlines (with lots of Ping-Pong balls)”

Here are some videos of the “Visualizing streak lines, path lines, and streamlines (with lots of Ping-Pong balls)” demonstration. The videos do not show the follow up drawings. The headings are links to higher resolution videos. Thanks to Scott Black of the Glenn Department of Civil Engineering for help with the videos and all the technicians in the department for help in building this and other demonstrations. Also, don’t forget to follow @nbkaye on twitter to get updates on this blog.

Path line (this GIF is a little unclear. I recommend the full video)

pathline

Streak line

streakline

Streamline

streamline

Visualizing streak lines, path lines, and streamlines (with lots of Ping-Pong balls)

The difference between path lines, streak lines and streamlines is often hard to visualize. There is a great video illustrating the differences between path lines and streak lines for an unsteady flow but the flow gets a little complex at times. I use a larger scale demonstration with Ping-Pong balls acting as the fluid particles. This one takes a little more equipment and a fair bit of clean up.

Equipment

  1. Lots of Ping-Pong balls (50+)
  2. A bucket
  3. A Ping-Ping ball gun capable of shooting 10 or so balls at a time.
  4. An air compressor

The Ping-Pong ball gun can be made from a four foot length of 1½” PVC pipe capped at one end. Attach a quarter turn valve to the cap and a compressed air connector to the valve (see picture below). You will need either a compressed air line or a portable air compressor in the class room to fire the gun.

PPG

Demonstration

Path lines (the trajectory of an individual particle)

Write the definition of a path line on the board. Load a single ball into the gun. Ask the class to take a mental image of where the ball is at each of the times that you call out during flight. Fire the ball across the front of the class room in front of the white board (use orange balls if you have a white board or white balls if you have a chalk board). While the ball is in flight, quickly count out loud from 1 up to say 5 or 6 (the actual number is not important). After the flight draw a series of circles on the board (one for each number you counted) roughly following the arc of the ball’s flight. Label them as (t=1), (t=2), … with (t=1) being the circle nearest the outlet of the gun. The labels refer to the locations of the ball at the times called out. Draw a line through circles to form the path line.

Slide3

Streak line (a line connecting all the particles that have passed through a single point)

Write the definition of a streak line on the board. Load about 10 balls into the gun. Ask the class to take a mental image of where all the balls are when you call out ‘NOW’ during flight. Fire the balls across the front of the board and call ‘NOW’ when the balls are in the air. After they have landed draw up a series of circles in an arc across the board. Label them (P1), (P2), … with (P1) being the particle that came out of the gun first. The labels indicate the order in which they left the gun. Draw a line connecting the circles to form the streak line.

Slide4

Streamline (a line that is instantaneously tangent to the velocity vectors of the flow)

Write the definition of a streamline on the board. Tell the class that you are going to throw a bucket full of balls across the front of the room and ask them to take a mental image of where each ball is and what its velocity is when you call “NOW”. Throw the balls across the front of the room and call out “NOW” when the balls are in the middle of the board. Once they have landed, draw a whole series of circles all over the board with arrows for the velocity vectors. Draw the vectors such that you can draw lines tangent to the arrows passing through multiple balls. I typically draw up about 20-30 circles. I often ask the class if the drawing is accurate. Then draw a series of streamlines that pass through the circles and are tangent to the velocity vectors.

Slide5

At the end of the class I ask each student to pick up a couple of balls each to help tidy up.

Discussion

The demonstration usually gets a good laugh, particularly when 100+ balls get thrown across the front of the room (I think the students don’t actually believe I am going to throw the whole bucket full). However, it does illustrate that a path line is formed over time (the counting during flight) as a single particle moves through space, whereas the other two lines are snap shots in time (when you shout “NOW”) connecting multiple particles.

Videos to follow shortly.