The conservation of energy, which is so innocently applied in elementary applications, can be actually quite complicated when the system in question has mass flowing into or out of it. One of the most popular textbook questions for provoking thought on this topic is the conveyor belt.
Two well-known texts, ‘Mechanics’ 3rd ed., by Symon and ‘Physics’ by Halliday and Resnick, cover the conveyor belt problem, although in slightly different ways. Both start from a common setup of a belt moving at a constant velocity $$\vec v$$ onto which mass is dropped at a rate $$r = \frac{d m}{d t}$$ from a hopper above. The motor powering the conveyor applies a varying force $$\vec F$$ so that the constant velocity is maintained as the material mass $$m(t)$$ grows. For completeness, the mass of the belt is assigned a value $$M$$. Since the problem is one-dimensional, the explicit vector notation will be dropped in what follows.
The derivation starts with the mechanical momentum of the belt, defined as
\[ p(t) = [m(t) + M] v \; .\]
The time-varying motor force needed to maintain the constant velocity has to match the change in the momentum and thus
\[ F(t) = \frac{dp}{dt} = \frac{dm(t)}{dt} v = rv \; .\]
The power supplied by the motor is
\[ {\mathcal P} = F(t) v = v^2 \frac{dm(t)}{dt} = v^2 r \; , \]
which can be manipulated into a more familiar form since both $$v$$ and $$M$$ are constant, to yield
\[ {\mathcal P} = \frac{d}{dt}\left( m v^2 \right) = \frac{d}{dt} \left[ (m+M) v^2 \right] = 2 \frac{d}{dt} \left[ KE_{sys} \right ] \; . \]
In words, the motor supplies power that is twice is large as the change in the kinetic energy of the system. Halliday and Resnick also derive this two-to-one relationship. Both texts then ask where the excess half of the power going?
Of course, these authors expect that the student would infer where the energy has gone. His argument would start with the notion that the law of conservation of energy is cherished and believed to be correct in all cases. If half the supplied energy is easily found in the form of kinetic energy of the belt plus material the rest must be ‘hidden’ in a less obvious place. This missing other half must have been converted into some other form. Continuing on in this vein, the student would then conclude that a force is needed to accelerate each bit of mass introduced onto the belt and that that force can only be due to friction between the belt and the material so introduced. Once friction is introduced, it is a short and easy step to conclude that the extra energy is converted into the internal energy of the belt or the material.
Since this is the obvious path for such a venerable problem, it is hard to believe that there is any controversy surrounding this conclusion. But as Mu-Shiang Wu points out in ‘Note on a Conveyor-belt problem’, The Physics Teacher (TPT) 23, 220 (1986), the student is often puzzled why the missing energy happens to be half of the supplied power. By his presentation in that article, Wu also implies that the student is not satisfied with the broad and general conclusion that missing half is dumped into heat, he also wants to see the explicit mechanism. Wu’s argument to explain this mechanism goes something like this.
Follow a bit of mass $$\delta m$$ as it falls from the hopper. Relative to an observer riding along the belt, this chunk is moving with velocity $$-v$$. Since this bit of mass must come to rest with respect to the belt over some period of time $$\delta t$$ there must a be a frictional force $$F_f$$ that arrests the mass’s leftward motion. Wu posits the form of this frictional force to be
\[ F_f = \mu N = \mu \delta m g \; ,\]
where $$\mu$$ is the coefficient of kinetic (or sliding) friction, and $$N$$ is the normal force supplied by the belt. This kinetic friction force results in a constant acceleration $$a = \mu g$$. Using the standard the kinematic relations for constant acceleration
\[ x_f = x_i + v_i \delta t + 1/2 a \delta t^2 \]
and
\[ v_f = v_i + a \delta t\]
and eliminating the time $$\delta t$$ it takes for the friction force to bring the chunk of mass to a stop (i.e. $$v_f = 0$$) leads to the total distance traveled as
\[ D = \frac{v^2}{2 a} \; .\]
The work done by the frictional force is then
\[ \delta W = F_f D = \frac{1}{2} \delta m v^2 \; , \]
and the power exerted is
\[ {\mathcal P_f} = \frac{\delta W}{\delta t} = \frac{1}{2} \frac{\delta m}{\delta t} v^2 = \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) \; .\]
At this point, Wu stops with the statement
Overall, I am suspicious of Wu’s argument. It has the attractive feature of having a explicit mechanism for the force that brings the material to rest on the conveyor belt but the use of the co-moving frame is done using a bit of a cheat. To demonstrate the roots of my suspicion, let me modify Wu’s argument starting just after the use of the constant acceleration kinematic equations. The total displacement (not distance) traveled is
\[ D = (x_f – x_i) = -\frac{v^2}{2 a} \; .\]
The work done by the frictional force is then (note the sign)
\[ \delta W = \int_{x_i}^{x_f} F_f dx = F_f (x_f – x_0) = -\frac{1}{2} \delta m v^2 \; .\]
Thus the chunk of mass loses energy in this frame and the power loss is
\[ \frac{\delta W}{\delta t} = -\frac{1}{2} \frac{\delta m}{\delta t} v^2 \]
or (taking the limit in the usual casual physicist style)
\[ {\mathcal P_f} = – \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) \; .\]
The power lost by the object is exactly one half of the total power supplied by the belt. This loss is assumed to go into heat so the energy balance is satisfied in a hand-waving way but there is this pesky problem associated with the two different frames. So I don’t think the puzzle is satisfied.
In the intervening years (1987-1990) after Wu’s initial article was published a number of other author’s published notes, critiques, and alternative ways of thinking about the conveyor belt problem. Judging by the different points-of-view expressed, the original unanswered question by Symon and Halliday and Resnick seems to have assumed a manifest truth that is not as obvious once one digs in as it is on the surface. Almost none of the arguments I’ve read in TPT have swayed me except for a letter by Marcel Alonso in response to Wu’s original article.
Next week I’ll cover Alonso’s argument and some details about variable mass systems.