Added Mass and Drag

How do you politely ask water to get out of your way?

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Added Mass and Drag

This lesson picks up where we left off in the lesson on Force Balance. If it’s been a while since you did that lesson, we recommend reviewing it before diving in here. Once you’re ready, and have refreshed yourself on weight and buoyancy, pull out your ellipsoidal BLIMP.

Newton’s second law can be written as the sum of the forces equals the mass of the system times its acceleration: ΣF=ma.  Let’s put this to work here.

• We know all the forces acting on the hull, right?  We had found the total vertical force to be ΣF=B-W_B-W_H=______N directed upwards (that is to say, this is a force vector with magnitude and direction).  
• We also know the weight of the helium filled hull; it is the weight of the hull plus the weight of the helium: W_B+W_H=_____N.  The mass of the helium filled hull is then this weight divided by the acceleration due to gravity, 9.81 m/s².  So, the mass m=______kg.
• You know the sum of the forces on the hull, and you know the mass of the hull.  Using ΣF=ma, what do you predict the acceleration of the hull to be if you let go of it?  a_predicted=_____m/s²

(Notice, I am using a subscript of “predicted” here to distinguish this from the acceleration we are going to measure in the next step).

To check our work, we can do an experiment to measure the acceleration of the hull when released between the floor and the ceiling of the room you’re in.  (Tie a string to your blimp, or you might need a ladder to fetch the hull.)

The relationship between distance traveled, s, initial velocity, V_o, acceleration, a, and time, t is: s=V_ot+1/2at².  If the initial velocity is 0 for something that begins at rest, this simplifies to s=1/2at².  So, to measure a, we need to measure the distance the hull travels and time how long it takes to make its journey.

• What is the distance from the floor of the room you are in to the ceiling? D=_____ m
• Subtract from that twice the radius of the hull, r (r=a, b, or c depending on which orientation you want to release your ellipsoid in).  We’re going to measure the acceleration of the hull from the floor to the ceiling, but the center of the hull is moving from one hull radius above the floor, to one hull radius below the ceiling: s=D-2r=_____ m
• Now let’s time how long it takes the hull to rise from the floor to the ceiling.  Make sure the hull is starting from a stand still.  Repeat that a few times until you are confident in your time measurement.  t=____ s

What then was the hull’s acceleration a=2s/t²?  a_measured=____ m/s²

Does this agree with what you were expecting with a_predicted?  

I venture to guess that your measured and predicted values are close, but not exactly in agreement.  Part of this may come from your measurement of distance or timing.  But part of it is also due to both drag and added mass.  Drag opposes the direction of motion for a body moving in a fluid, and is proportional to velocity squared.  Added mass comes about because a body accelerating through a fluid has to accelerate the fluid it is displacing.  This effect is often negligibly small on vehicles like airplanes, but we can in fact see its effect with your ellipsoidal hull!  If you’re interested in doing a controlled experiment to measure drag and added mass, check out: http://salt.uaa.alaska.edu/jim/mass.pdf.  

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Next Steps

You’ve demonstrated excellent technical knowledge getting through this lesson. Develop your technical communication skills with the next section on the Literate Engineer.

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Last updated: November 23, 2022.