NCU Physics Demonstration Lab 國立中央大學物理演示實驗

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1. Murphy’s Law is typically known by “if something can go wrong, it will”. When applying the Law to the case of toast fall, it denotes that, with the tendency of turning bad to worse, the tumbling toast always lands jam side down. As shown in the demonstration video, the tumbling toast lands on the side with a cross (or the side of jam). However, unsurprisingly, the landing dynamics has nothing to do with the Law; it is a complicated question involving factors like the size of toast, the onset tumbling angle, the torque of toast rotating off the table, and the moment of inertia of the toast.
2. Neglecting the frictional force, the free body diagram of the toast at the initial state is shown in the figure below:


Based on the law of Conservation of Energy, the Potential Energy gained from table height is transformed to Kinetic Energy of center of mass and Energy of rotation during the fall. In the figure above, is the angular velocity of toast rotation; is the angle included by the toast and the surface of table; is the length of toast; , the velocity of center of mass, can be divided to vertical to the toast and parallel to the toast; )， is the mass of toast; is the moment of inertia. Equate the Potential Energy and the transformed Kinetic Energy, we obtain
…………(1)
Since

and the toast’s moment of inertia is
………………………………..(2)
(1) can be rewritten as

and further simplified to
………………………………..(3)
From the Impulse-Momentum Theorem, (3) can be represented as

where is the time for the toast’s disconnecting from the table, which is considerably short.
The torque formed by the distance r, from the toast’s center of mass to the point of contact on the table, is:
……………………………………………(4)
The r here is a function of time, , leading to the average of r:
……………………(5)
In the very beginning of the toast’s disconnecting from the table, . Based on , fit the result of (5) into (4):
………………………(6)
From the Parallel Axis Theorem, the toast’s moment of inertia is obtained:

……………………………..(7)
Fit the result of (2) and (7) into (6), we have:

simplified to
………………………(8)
Insert (8) in (1), we obtain

and derive after calculation

In our demonstration, given the included angle , , the length of the toast , ，
and the height of the table ，it takes only 0.4 seconds for the toast to fall and land. The overall angle of rotation, containing the initial tilt angle , , and the tumbling angle , ，an angle between and ) In other words, the toast turns upside down and lands on the side with a cross (jam).