By Michelle Lu

Conservation of angular momentum: how cats almost always
land on their feet

Cats have amazing flexibility in their backbone, allowing
them to separate their bodies into 2 rotational axes – the front half and the
back half.

By the conservation of angular momentum:

L

_{1 }= L_{2}
I

_{1}w_{1}= I_{2}w_{2}
Cats have the ability to increase or decrease their moment
of inertia of their front or back halves by spreading out their legs or
retracting their legs. If a cat is
dropped with its stomach facing upward, first the cat would increase the moment
of inertia in the back, extending its back legs, and decrease the moment of
inertia in the front, retracting its front legs. This allows the cat to
increase its angular velocity in the front and decrease its angular velocity in
the back, and the front half of the cat will twist quickly while the back half
moves very little. Next, the cat will stretch out its legs in the front,
increasing its I in the front and decreasing its I in the back. The back of the
cat will rotate quickly while the front of the cat will move very little, and
the entire cat would have rotated 180 degrees. All four legs of the cat would
be facing the ground and the cat is prepared to land.

Since the cat splits its body into two rotational axes,
let’s assume each half of the cat is a long uniform rod and I = (1/3) ML

^{2}_{. }Assume the cat is 5 m long, and L_{1}= L_{2}= 2.5 m when the cat is neither extending nor retracting its legs. Also assume that the cat retracts or extends its legs by a length of 0.5 m.
I

_{1}w_{1}= I_{2}w_{2}
(1/3) ML

^{2}_{1 }w_{1}= (1/3) ML^{2}_{2}w_{2}
L

^{2}_{1 }w_{1 }= L^{2}_{2}w_{2}
(2.0)

^{2}w_{1 }= (3.0)^{2}w_{2}
w

_{1}/w_{2}= 2.25
From this ratio, we see that in the beginning when the cat
only twists its front half of the body, the front of the cat rotates almost
twice as fast as the back rotates.

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