The concept of added mass can be a useful but perhaps a little hard to visualize (either the added mass or the impact of added mass on a system). This is a simple demonstration adapted from Granger’s Experiments in Fluids to illustrate the impact. The basic idea is that a solid accelerating in a fluid will accelerate fluid in its wake effectively increasing its inertia. The method for demonstrating this is to use a spring-mass system that is released and oscillates in air and then water. The added mass in water is much larger than that in air, increases the system inertia, and reduces the frequency of oscillation.

**Equipment**

- Container of water
- Spring
- Mass
- Support structure
- Stop watch

**Demonstration**

- Mount the mass on the spring in air, extend the spring, and release.
- Measure the time taken for the mass to oscillate 20 times (may be more or less depending on the losses in the system and frequency of oscillation).
- Mount the mass on the spring and place it in the water container (ideally only the mass on a long cord with the spring dry), extend the spring and release.
- Measure the time taken for the mass to oscillate 20 times (this may be harder and you may have to have fewer oscillations this time as the water damps out the motion fairly quickly).

**Analysis**

The frequency of oscillation (*f*) of the system is

*f~(k/m)*^{1/2}

where *k* is the spring constant and m is the mass. There are a number of methods for getting to this result. Dimensional analysis is an easy method where the dimensions of the parameters are [*f*]=T^{-1}, [*k*]=MT^{-2}, and [*m*]=M. Alternatively one can write down the equation of motion

*m(d*^{2}x/dt^{2})=-kx

and solve to get

*x=x*_{0}cos((k/m)^{1/2}t+Θ*)*

giving an oscillation frequency of

*f=(k/m)*^{1/2}/2π*.*

Regardless of how you get there, the main result is that the frequency of oscillation is inversely proportional to the square root of the mass, i.e.

*(2π**f)*^{2}=k/m.

When the mass is immersed in the water the effective mass of the system increases due to the added mass. Therefore, the Frequency of oscillation of the system will also drop. Denoting the added mass by m_{a} the frequency in water (*f*_{w}) is given by

*(2π**f*_{w})^{2}=k/(m+m_{a}).

For a known mass, the spring constant can be calculated from the frequency of oscillation in air. The added mass can then be calculated from this data and the frequency of the system in water.

I ran a test using the equipment shown above. In this case the mass was 0.15 kg and the frequency in air was *f*=1.6 hz giving a spring constant of

*k=(2π**f)*^{2}m=15 N/m.

The frequency of the system when immersed was *f*_{w}=1.45 hz giving and added mass of

*m*_{a}=(k/(2π*f*_{w})^{2})-m=0.031 kg

or 20% of the original mass.

An index of all the demonstrations posted on this blog can be found here. 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.

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