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:
L1 = L2
I1w1 = I2w2
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) ML2. Assume the cat is 5 m long, and L1 = L2 = 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.
I1w1 = I2w2
(1/3) ML21 w1 = (1/3) ML22w2
L21 w1 = L22w2
(2.0)2 w1 = (3.0)2 w2
w1/w2 = 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.