3. How Toast Tumbles

Before proceeding down the slippery slope of calculations, we should consider the simple forces behind Buttered Toast Ballistics. Here, we’ll list the forces that act on the Toast and get a physical feel for why Toast Tumbles (flips end-over-end) when dropped.


Figure 1a depicts the situation described previously as The Initial Conditions. To be explicit these are The Initial Conditions: The Toast starts on the Table Ledge, or some other surface about the same height, but we'll restrict that height to be between the surface of a low Table and the mouth of a tall person, standing. It is reasonable to assume most Toast spends most of its life between such altitude extremes. The Toast starts off with no velocity, thus we assume it has be placed clumsily, but not violently, in such a way it will topple and fall primarily under the influence of Gravity. The preferred orientation most people handle their toast is also Butter Up.

First, let's consider why Toast falls. This is easily understood by considering its placement on the ledge of the Table, with its center of mass beyond the ledge, as marked by the x in Figure 1a. Only two forces are relevant. Gravity is pulling down on the Toast (F_g, blue arrow) and the Table responds with the normal force (F_n, blue arrow). If the center of mass does not extend over the ledge (Figure 1b) then the Table supports the Toast, and the Toast does not move. Physically, the rigidity of the Table provides F_n equal and opposite F_g. More colloquially we’d say you successfully placed your Toast on the Table!


Many people find it intuitive that the Toast will rotate if the Table does not support its center of mass, as in Figure 1a. To see this explicitly, consider the red dots in Figure 1. These dots mark the Table Ledge, the axis around which the Toast will rotate. In Figure 1b, more of the Toast’s mass lies on the right of the red dot than the left, so it would try to rotate clockwise. However, the Table supports the right hand side, so the Toast just sits there. In Figure 1a more of the Toast is on the left of the red dot, so the Toast tries to rotate counterclockwise. There is nothing to support the left side of the Toast and we soon end up in the situation in Figure 1c, with the Toast rotating and angled as it tumbles off the Table.


The same two forces are still present, gravity and the Table’s rigidity. The situation in Figure 1c is more complicated because more things are happening. Focusing on gravity, we notice that it does two things to the Angled Toast. Even as the Toast rotates, F_g helps it rotate faster by pulling at a right angle to the line connecting Toast’s center of mass to the pivot point at the Table Ledge. (This is caused by the torque τ = F_g x d. People who have studied physics previously will recognize that it is now necessary to consider the vector nature of forces to quantify what is going on.) For this discussion, we notice that only a component of F_g torques the Toast and makes it spin faster, so we label this portion F_t. The other component of F_g is pulling along the length Toast, causing it to slide off the Table, so we label it F_s. Once the Toast begins to tilt, we distinguish these two components of gravity (the green arrows in Figure 1c) according to what type of motion they produce: making the Toast tilt faster (F_t), and causing sliding off the Table (F_s). (Notice that these are not new forces, but a new way to describe the forces present, based on the type of motion they produce. Gravity always pulls the Toast straight down. What is changing is the geometry of the Toast-Table system, so gravity’s tug moves the Toast in a slightly different way.)


The Table Ledge also behaves a bit differently. F_n is still present, but we now must add a force of friction, F_f, which opposes the sliding of the Toast. The black arrow in Figure 1c represents this force, which was not relevant in Figure 1a and 1b until we began considering sliding and pondered the force that opposed it.


All these forces act in concert; by the time the Toast leaves the Table, there will be several types of motion. Most importantly for the Buttered Side Down discussion, the Toast will be tumbling end over end. However, the sliding will likely cause some lateral motion (to the left), and it will already be falling toward the floor when it leaves the Table. Importantly, once the Toast leaves the Table it will be in free fall; it will accelerate vertically, but its tumbling and lateral motions will be constant without the influence of the Table. Thus, whatever conditions we calculate for the Toast as it leaves the Table will be easily extended to the time at which it hits the floor. All the hard work will be performed in calculating what is going on with the Toast before it leaves the Table.


A purist might say that Toast will experience drag (friction from the air) as it falls, which will change its lateral and tumbling motion. Although I will directly confront this at some point down the road, drag forces should be fairly negligible at the typical speeds that Toast moves. There are several other forces present that I have not mentioned yet, including the Coriolis effect in Earth’s rotating reference frame, and tidal forces from the moon and sun. Clearly, not all of these are important. Instead of trying to include EVERYTHING at once, it is better to come up with a simple model and then add complications and see which ones are important and what is needed to make theory and experiment agree. In fact, we’ll start even simpler, without including all of the effects included in this discussion, as it is useful to see what happens in the absence of sliding.