What Makes a Locomotive Move?
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That seems like a fairly stupid question at first. But given that stupidity denotes the inability or the unwillingness to discern the obvious, we may well ask whether that question has a truly obvious answer. While lying in bed one night I tried to diagram in my mind the forces acting on a steam locomotive and failed to find the one that actually moves the locomotive forward on the rails. Not such a stupid question after all, it seems. So how shall we answer it?
We can start with a simple, but not trivial, example and ask How does a mining cart move? In small mines the miners push carts on paired rails that we might call a railpath rather than a railroad. By rolling the carts on rails the miners minimize the carts' resistence to movement. But, still, someone must exert a force on a cart to make it move on those rails.
Oversimplifying considerably, as we do in physics, we represent a miner as a long straight bar leaning against the rear of a cart. As in the case of a ladder leaning against a wall, the primary force acting on the bar, its weight, comes from the action of gravity. We can treat that force as if it acted solely and entirely at the bar's center of mass. Because we have the bar tilted, that force exerts a torque about the point where the bar touches the ground. Where the bar touches the cart that torque manifests itself in a force that breaks into two components B a vertical force, which the cart transmits to the rails and thence to the ground, and an horizontal force, to which the cart responds by moving. Newton's third law of motion necessitates that the cart's inertial reaction to that force exert an opposing force of equal magnitude upon the bar, basically through a slight compression of the cart's material to convert the inertial reaction force into an elastic force. That equal and oppositely directed reaction gets transmitted through the bar, by way of the internal forces that make the bar rigid, to the point where the bar touches the ground. Friction between the bar (the soles of the miner's shoes) and the ground mediates the transmission of the force to the ground and prevents the bar from slipping. The ground reacts to that applied force in the same way in which it reacts to the bar's weight, by compressing slightly to generate an elastic force.
Thus, all of the forces exerted in that scenario balance perfectly. That fact remains true to Reality, even though the forces exerted by the cart change in their nature. If we lock the cart's wheels so that they cannot turn, then the cart draws the force with which it pushes back on the miner from the static force of elastic compression that the wheels get through their frictional contact with the rails. When we unlock the wheels and let them turn freely, that static force relaxes to zero as the inertial force due to the cart's acceleration (the famous F=ma of freely accelerating bodies in the basic physics course) takes over the reaction to the miner's push.
When the cart begins to move, the miner will tend to topple forward and, if he takes no remedial action, he will fall flat on the track, succeeding only in moving the cart a short distance. The miner avoids that fate by stepping forward as the cart moves away from him, thereby maintaining his original tilt. In that way a third body (the miner) coming between the cart and the track acts as an intermediary for the forces that make the cart and the track move relative to each other.
We have no such intermediary in the case of a steam locomotive. So how does the interaction between the locomotive and the track produce an horizontal force that not only moves the locomotive forward, but also enables it to pull a train of carriages with it? To aid in our analysis of that question let's imagine that we have before us a simplified version of the Stourbridge Lion, which was built in England and brought to the United States in 1829.
I have chosen to use the Lion as my example of a steam locomotive because it drew its mechanical power from vertically moving pistons. That arrangement eliminates one of the sources of my confusion in my initial analysis and makes the Lion the perfect vehicle for exploring the weird realm where statics segues into dynamics. Rigid rods connected the Lion's two pistons to a pair of walking beams that pivoted on a support that held them above the Lion's boiler. Long drive rods descended to connect to the locomotive's large rear wheels, coming vertically down from pivots between that support and the points where the piston rods connected to the walking beams. A connecting rod ran from each of the rear wheels to one of the equally large front wheels so that all four wheels would generate thrust. But that configuration does not allow us to assume that the force exerted by the pistons gets directly applied to accelerating the locomotive.
So we have brought our version of the Lion into the roundhouse and suspended it in a harness hanging from an overhead crane. If we let steam pass from the boiler to the cylinders, the pistons will move, the walking beams will rock up and down, and the drive rods will make the wheels turn on their axles. The reciprocating motion of the pistons, transferred to the drive rods through the walking beams, makes the wheels turn by exerting a force on the off-center crank pin on one of each wheel's spokes. By virtue of its location some distance from the axle, each crank pin converts the redirected thrust of its associated piston into a torque that turns the wheel.
Now imagine that we have turned our version of the Lion upside down in a frame that leaves it perfectly free to move horizontally. The real Stourbridge Lion would not have worked upside down, of course, but we can pretend that our version will. For this exercise we take a length of railroad track, turn it upside down, and lay it on the Lion's wheels. If we now let steam into the Lion's cylinders, the wheels will turn and the track will go in one direction while the Lion, in Newtonian equal-and-opposite reaction, moves in the opposite direction. We certainly expect that to happen, but how did it happen? How do vertical forces get transformed into horizontal forces?
We need to remind ourselves here of Newton's first law of motion - a body remains in a given state of motion until an unbalanced force acts upon it. More specifically we can say that at any instant of time, a body moves in response to only the forces exerted upon it at that instant. Equivalently, we could say that a body reacts to the forces that it exerts at the instant that it exerts them, thereby reflecting the first law through the third law.
What that means for us comes to the wheel exerting a force on the track only where and when it touches the track. As each piston exerts force on it through the walking beam, each drive rod exerts upon its wheel a force that becomes a torque. The forces that hold the wheel's atoms together and give the wheel its rigidity transfer the torque around the wheel, acting to make the wheel rotate. Where the wheel meets resistance to rotation, through frictional contact with the rail, the metal deforms slightly to create an elastic force that pushes on the rail. Thus we can account for the force that accelerates the rail and its associated track.
That the track exerts an equal and oppositely directed force upon the wheels we do not doubt. The metal of the rails deforms slightly and thereby exerts an elastic force that pushes back on the wheels as the wheels push on the rails. So the accelerating track exerts a force on the rims of the wheels as the wheels turn. But if we turn our Lion right-side up again and replace it on ground-anchored rails, that force will not make the Lion move. The force exerted between the rim of a wheel and the rail has more in common with the static force that keeps the feet of a ladder from sliding on a sidewalk. But, we notice, friction makes the wheel-rail contact point a pivot about which the torque in the wheel might act.
A torque put into a wheel can produce direct forces by way of the principle of the lever. To illustrate that principle we imagine placing a long rigid bar on a fulcrum such that one part of the bar hangs over one side of the fulcrum and the other part hangs over the other side of the fulcrum. We apply a force to one end of the bar to create a torque about the fulcrum and that torque passes to the other side of the bar, where the torque can then manifest a direct force on that side of the bar through deformations of the bar's metal. That means that when we apply a force to one end of the bar, the upper part of the bar stretches slightly and the lower part undergoes a slight compression, creating at each place along the length of the bar a small torque about the bar's centerline. Those torques add up and at the opposite end of the bar from where we push those torques generate a force that, combined with the distance from that end to the fulcrum, mimics the torque. If we lock one end of the bar in place, then a force applied to the other end gets manifested as a force that tends to move the fulcrum. This remains true to mechanics even if we bend the bar where it meets the fulcrum. All we need to apply the principle of the lever is some means to convert an applied force into a torque and a means to make that torque exert a force.
So in the Lion the thrust of a drive rod creates a torque in one of the wheels. With the Lion's axle as the fulcrum, that torque exerts a thrust upon the rail. With the point of wheel-rail contact as a fulcrum, that torque exerts an equal and oppositely directed thrust upon the axle and, through it, upon the Lion's frame. That makes the Lion move.
Now we know that when the engineer lets steam into the cylinders the drive rods will convert the thrust of the pistons into torque on the wheels. Because each drive wheel has two points that can serve as fulcrums the torque in the wheel comes manifest as a force couple, one element of the pair exerting thrust against the rail and the other exerting an equal and oppositely directed thrust against the Lion's axle. If we connect a train of carriages to the Lion, we can transport an amazing quantity of passengers and freight (by 1830's standards) down the track at breathtaking speeds as high as twenty miles per hour!
It makes no difference how we produce the torque on the drive wheels. We can mount the cylinders horizontally at the front of the locomotive rather than vertically at the rear, thereby creating what we normally think of as a proper steam-driven locomotive. Or we can make the axle of a pair of drive wheels the rotor of an electric motor, as we do in the modern diesel-electric locomotive.
Thus we have answered the question with which I titled this essay. It may seem strange to spend so much effort to address such a small trivium in the most basic Newtonian physics. I may seem to have devoted too many words and only words (where, after all, did I leave the equations?) to a completely frivolous topic. But I have a sound reason behind my apparent madness.
We have a statement called Cobbett's Rule: "I speak not only so that I can be understood, but so that I cannot be misunderstood." In accordance with that rule physicists must sometimes pump impressive numbers of words into what look like very small concepts in order to clarify them before applying the relevant equations. We find that practice less necessary in experimental work, because the empirical-inductive Scientific Method is self-correcting, but it becomes essential to any effort to create an axiomatic-deductive system like the Map of Physics.
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