Physics behind the spillnot

(physics is so cool!)

The cool physics of the SpillNot is that the liquid does not spill or slosh in the cup and the cup does not tend to slide on the SpillNot as it’s carried, even if it is swung all around. Why not? The answer is that the loop handle (let’s think of it as a ‘thread’) is very flexible and cannot deliver a lateral acceleration to the cup (because it has no ‘sheer strength’), which is what usually makes liquid spill from an open container that is not tipped.




When people spill coffee while walking with an open mug, the liquid ‘sloshes’ out due to a horizontal acceleration of the cup. When the cup is accelerated, it applies a horizontal force to the liquid. The liquid, of course, has no fixed shape so the horizontal force causes a build-up of liquid at the side of the cup that is applying the force. If the build-up is higher than the side of the cup, there is a spill. If the force ceases without a spill, gravity pulls the liquid back down and the momentum of this flow causes a smaller build-up of liquid on the other side of the cup. With no additional forces, these oscillations eventually die down and equilibrium is reached with the surface of the liquid flat and level.

Interestingly, when walking with a cup of liquid, a person may inadvertently apply horizontal accelerations to the cup at an interval that matches the resonant frequency of the oscillations described above. When this occurs, the amplitude of the oscillations can increase, causing the liquid to spill even if the person carrying the cup has not applied any single particularly large horizontal acceleration to the cup. (I first posted this observation in 2011, and in 2012 a study was published by researchers at UCSB that confirmed this observation with copious scientific evidence. A more general article was written about the physics of the SpillNot in 2014.)

What happens when a person applies a horizontal acceleration to the handle (upper end of the ‘thread’) of the SpillNot? Because the thread is very flexible, no lateral force is applied to the saucer and thus no lateral force is applied to the cup. Instead, as the top of the thread moves horizontally, the saucer and cup swing up slightly in the opposite direction and the thread exerts a force on the cup through the saucer causing acceleration in the only direction in which the thread can exert a force, toward the upper thread end that is held. The direction of this force is parallel to the sides of the cup (by the SpillNot design), and thus does not tend to cause sloshing. At the same time, gravity pulls down equally on the saucer, cup, and liquid molecules in the cup, counter-acting the swing caused by horizontal acceleration of the upper thread end without causing sloshing.

More generally, swinging does not tend to cause the cup to slide off the saucer or liquid to slosh in the cup because the only force acting in a direction that is not parallel to the sides of the cup is the force of gravity, and gravity is equally accelerating all molecules of the saucer, cup, and liquid in the cup.

If the SpillNot is swung all the way around in a loop such that the cup is upside-down at the top of the arc, the liquid does not pour out of the cup due to the pseudo force, centrifugal force, which acts in the opposite direction as gravity when the cup is at the top of the arc.

As an aside, it turns out that for the webbing loop handle to lie flat and have no twists, it must be a Möbias strip (the strip is sewn together on the standard SpillNot handle for ease of use).


Forces/Vectors, Newton’s Law’s, and the SpillNot

Carrying an Open Beverage Directly by the Handle. First consider a walking person holding a cup by the handle. Let’s say the person inadvertently applies a horizontal force to the cup, by abruptly moving the handle of the cup. The cup is then accelerated to the left in the direction of the horizontal force. The normal force, Fn, which is exerted on the liquid in the cup in a direction perpendicular to wall of the cup is increased by the acceleration of the cup, causing the liquid to accelerate in the same direction. By Newton’s third law we know that in reaction to the force, Fn, the liquid exerts a force of equal magnitude but in the opposite direction, Fl. There is also the force of gravity (mass * the constant, g), Fmg, acting on the liquid in the downward direction. 

At rest, the surface of liquid in a container is perpendicular to the direction of the force of gravity, Fmg, as is depicted in the above drawing. However, if we consider the reference frame of the cup during acceleration, the liquid is subject to two forces, Fmg and Fl, which act in combination. To account for both these forces, we can compute the net force, Fnet, by vector addition. If we align the two vectors, Fmg and Fl, tail to head, then the net force is the vector, Fnet, which has length and direction extending from the tail of the first vector to the head of the second. In reaction to the acceleration, there is a realignment of the surface of the liquid in the cup to be perpendicular to Fnet, rather than Fmg, as depicted above in the drawing. The larger the force Fl, the more tipped the surface will be while the cup is accelerating. If the surface is sufficiently tipped, the side of the cup will not be tall enough to retain the liquid, resulting in a spill.

Carrying an Open Beverage with the SpillNot. Now suppose an open beverage is being carried on a SpillNot rather than directly by the cup handle. Instead of abruptly moving the handle of the cup, the person abruptly applies a horizontal force, Fh, to the top part of the SpillNot handle.

Because the SpillNot handle is flexible webbing, it is very strong in terms of its tensile strength, ability to pull or exert a force in the direction of Fn, above, but it cannot exert a lateral force to the tray or cup. Thus when the force, Fh, is applied to the handle and it begins to accelerate, the cup and tray don’t move much, resulting in a tilt to the whole system (as depicted in the drawing above).
The force of gravity, Fmg, and the normal force, Fn, are acting on both the cup and liquid within the cup. The net force, Fnet, is in the same direction as the force acting on the handle, Fh, and will result in equal acceleration to the tray, cup, and liquid within the cup. Therefore, in the reference frame of the cup there will be no lateral forces on the liquid to cause spillage.

Actually, the system is a bit more complicated because we have not accounted for torque, which is the rotational acceleration of the cup and liquid. However, in a typical situation of a person walking with an open beverage on the SpillNot, the effects of torque will be minimal.