The Inclined Plane on Wikipedia

For those who live under a giant rock: Wikipedia is a user-created web-based encyclopedia. In conversation, I usually refer to it as a compendium of “common knowledge,” realizing of course that many elements of common knowledge are not common, and that many elements are flat wrong. Case in point: the Wikipedia entry on the inclined plane. (I refer to the article as it appeared on Sunday afternoon, 28 October 2007. The section on which I comment is reproduced in its entirety below, including the figure and text.)

The inclined plane is treated in detail in virtually every first-year physics course on the planet. I suspect that nearly everyone who took physics in college remembers it. We use it to teach students the quantitative aspects of Newton’s second law in two-dimensions, and to introduce friction. The Wikipedia article, although somewhat brief, touches on both of these. If I were grading the entry as an answer to a exam question, it would earn 3 out of 5 points, meaning that it gets some things correct, but that it also reveals major errors of fact or reasoning unacceptable for a first-year physics student.

First, let’s look at the figure included with the article:

This is a free-body diagram showing all the forces acting on the box. It doesn’t include any of the forces acting on the inclined plane, which is the correct thing to do. But there are two errors here that professional physicists should immediately recognize. First, the mg vector should either be erased or marked out. Once the weight has been resolved into components (mgcosθ and mgsinθ) and those components drawn on the figure, the mg vector is superfluous. With respect to a rotated coordinate system that puts the x-axis parallel to the incline, there are only four forces acting on the box, not five. When students leave the mg vector on the figure, not erasing it or marking it out in some way, they are prone to including both mg and its vector components in calculations of Newton’s second law. I stress to my students that after any vector has been resolved into components, it should be marked out in a way that makes it clear that it should not be included in any calculations.

The second error is with the friction vector, f. It’s correct that the friction vector must point parallel to the plane. But there is nothing in the figure or the associated discussion (see below) to indicate whether the boxing is moving, and if so, in what direction. If the box is stationary in the figure, then the force vector f represents a static friction force, and points in the correct direction. If the box is moving down the plane, the force vector f represents a kinetic friction vector and again points in the correct direction. If the box is moving up the plane, the force vector f is again a kinetic friction vector, but it’s pointing in the wrong direction–in this case it should point down the plane. It’s an unacceptable error to not specify whether the box is moving, and if so, in what direction. To help clarify things further (no amount of clarification is too much in physics), a subscript should be included with f, f_s for static friction, and f_k for kinetic friction.

Now let’s examine the discussion that goes with the figure:

The inclined plane gives rise to a common elementary physics exercise. Consider an object placed on an inclined plane, and describe mathematically the forces acting upon that object. There are three forces acting on the body (neglecting air resistance):

1. The normal force (‘N’) exerted by the plane onto the body,
2. the force due to gravity (‘mg’ – acting vertically downwards) and
3. the frictional force (‘f’) acting parallel to the plane.

The gravitational force may be visualised as two components: A force parallel to the plane (‘mgSinθ’) and a force acting into the plane (‘mgCosθ’) which is equal and opposite to ‘N’. If the force acting parallel to the plane (‘mgSinθ’) is greater than the frictional force ‘f’ – then the body will slide down the inclined plane – otherwise it will remain stationary.

When the slope angle (‘θ’) is zero, sinθ is also zero so the body does not move.

I don’t think it’s quibbling to complain that point 1 does not specify the direction of the normal force N–always perpendicular to the plane; after all, points 2 and 3 take care to specify the directions of the weight mg and the friction force f. Why not include five words to make clear how we determine the direction of N? But like the figure, the real problem here is that the text is ambiguous regarding the friction. It does not specify whether it’s a force of static friction or a force of kinetic friction, although it appears the author had a static friction force in mind.

“If the force acting parallel to the plane (‘mgSinθ’) is greater than the frictional force ‘f’ – then the body will slide down the inclined plane – otherwise it will remain stationary.”

Because the type of friction force is not specified, this statement is wrong and misleading. If f were specified as a static friction force, this statement would be correct. The static friction force is a responsive force: it grows to resist relative motion between the box and plane, up to some maximum value that depends on the particular materials involved. Quantitatively, it’s expressed as f_s ≤ μ_s N, where μ_s is the coefficient of static friction. (The maximum value is given by f_s,max = μ_s N. ) The value of μ_s depends on what the box and plane are made out of. Suppose the box is initially stationary on the plane. If we increase the angle of the inclination, the component of the box’s weight that points down the incline, mgsinθ, increases until it overcomes the static friction force. At that point, the box begins to slide down the plane.

But if f is understood as a force of kinetic friction, then it isn’t necessary for mgsinθ to be greater than f_k for the box to slide down the plane. If mgsinθ and f_k are balanced, the box simply slides at constant speed. It is also possible for the box to slide down the plane if mgsinθ is less than f_k: it slides down the plane with decreasing speed, eventually coming to a stop. Finally, it is possible for the box to slide up the plane if given an initial shove, eventually coming to a stop if the plane is very long (and flying off the top otherwise), since mgsinθ and f_k both point down the plane.

When the slope angle (‘θ’) is zero, sinθ is also zero so the body does not move.

Notice that this statement constitutes an entire paragraph, so it apparently says all there is to say about the zero-angle case. Taken literally, it makes the amazing assertion that boxes do not move on flat surfaces, where “flat” is defined with respect to the local acceleration due to gravity. If the reader of my article will take a book off the shelf, and shove it across the floor, she will prove this statement incorrect. Certainly there are specific circumstances under which this statement describes the behavior of a real box. For example, an initially stationary box on an inclined plane will remain stationary if we decrease the inclination angle.

Does the author of this article understand the distinction between static and kinetic friction? Almost certainly, yes. But in assessing the article, or grading it as an answer to an exam question, I can’t make guesses about what the author may or may not understand. I have to deal with what the sentences say, not what they would say if I could step in and charitably make a few edits on the author’s behalf. The author of these words is not the problem, it’s the words themselves. This is why we stress clarity and specificity in physics to point that our students get sick of hearing it. Even in elementary physics problems such as the inclined plane, so many details must be drawn out unambiguously in order to make sure we say nothing wrong or misleading, that it becomes impossible to summarize things in only a few words. That’s why my article is much longer than the Wikipedia article, and even here I have not fully explored all we can say about the inclined plane.

As I examine the edit history of the Wikipedia entry, it appears the article has stood as I recorded it here for at least six days. Perhaps it will soon be improved.

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