Mounted at the top of the vertical rod in the photograph above is a permanent-magnet DC motor with a reducing gear box, to whose output shaft is attached a stepped wheel. Attached to the largest-diameter step on the wheel is a six-foot string with a loop at the free end, from which you can hang a mass. The terminals of the motor are connected to a 6.3-volt (#47) miniature lamp. If the string is not already wound around the wheel, wind it around the wheel, and turn the wheel so that the loop just touches it. Hang the 1-kg mass (shown) from the loop, and release it. The lamp lights brightly as the mass descends. If as the mass is descending, you disconnect one of the clips from the lamp, the mass suddenly and greatly accelerates. You can also hang either of the other masses shown (200 g and 500 g), to compare how they fall, and how brightly they light the lamp as they do so. Please do not yank on the string or jerk the wheel. This will produce excessive current, which will burn out the lamp.

Demonstrations 72.03 – EMF induced by moving magnet, and 72.06 – EMF induced by moving conductor, illustrate the production of a current in a coil via the induction of an electromotive force by the changing of the magnetic flux through the coil, which we can express by the following equation: E = -NdΦ_B/dt, where E is the induced EMF, N is the number of turns in the coil, and the derivative is the change in magnetic flux with time. This is the physical principle that forms the basis for generators that produce electric power. Demonstration 72.12 – AC/DC generator, illustrates the operation of such devices. One type of generator is essentially a permanent-magnet DC motor run in reverse. Normally, one applies a voltage across the motor to push a current through the armature coils, thus generating a torque on them through repulsion between the magnetic field set up by this current, and the magnetic field of permanent magnets, in which they sit. (See demonstration 68.50 – Permanent magnet DC motor.) If, instead, one applies a torque to the shaft of the motor to spin it, the motion of the armature coils through the magnetic field of the permanent magnets induces an EMF in them, which is proportional to the speed at which they rotate. If a load is connected between the terminals of the motor, this EMF produces a current in the armature coils and in the load. Thus, whereas by placing a voltage across a motor to cause it to turn one converts electrical energy to mechanical energy, by applying a torque to the shaft of the motor to turn it and generate an electric current in a load, one converts mechanical energy to electrical energy. In either case, one can use the converted energy to perform work.

In the falling weight generator, gravity accelerates the mass downward, producing tension in the string, which exerts a torque on the wheel. This torque is r × T, where r is the radius of the wheel and T is the tension in the string. The tension in the string is equal to the force exerted by gravity on the mass, minus its acceleration, or T = mg - ma. The torque exerted on the wheel by the falling mass causes a current to flow in the armature coils of the generator and in the lamp as described above. According to Lenz’s law, the current flowing in the armature coils produces a magnetic field that opposes the magnetic field of the permanent magnets. (See demonstration 72.09 – Lenz’s law.) This causes repulsion of the coils from the magnets in one direction, and attraction in the opposite direction, and thus a torque that opposes that exerted by the falling mass. For simplicity, we shall call this “repulsion” or “magnetic repulsion.” As the mass falls, this repulsion balances the force of gravity (according to the different radii at which the magnetic repulsion and the tension act about their respective shafts, and the ratio of the gear box, the combination of which affords the generator significant mechanical advantage relative to the string). T = mg, a = 0, and the mass falls at constant speed. How much current is produced at a particular speed depends on the size of the load. The larger the load (the smaller its resistance), the greater the current flow for a particular EMF (which, as noted above, is proportional to the speed at which the generator shaft rotates). The speed at which the mass falls depends on how much current is necessary to produce a strong enough magnetic field that the aforementioned repulsion balances gravity, which in turn depends on the mass, m. The greater the mass, the greater the speed at which it falls. As the resistance of the load increases, the speed at which a given mass must descend to generate enough current for the repulsion to balance gravity, increases. If we disconnect the load (resistance goes to infinity), then no current can flow, there is no magnetic repulsion, and a approaches g. The mass would be in free fall, except that friction and rotational inertia in the generator and gear box cause its acceleration to be less than g.

As noted above, the current for which the tension balances the force of gravity varies with the mass hanging from the wheel. If you hang the 200-g mass from the wheel, it falls slowly, but does not light the lamp. If you hang the 500-g mass, it falls faster than the 200-g mass, and it lights the lamp dimly. If you hang the 1-kg mass, it falls faster than both of the other masses, and it lights the lamp brightly. In all three cases, of course, if you disconnect the lamp (the load), the mass suddenly accelerates greatly.

This behavior makes sense, because it requires energy to cause a current to flow (1 V = 1 N·m/C, or 1 J/C). Gravity provides this energy by exerting a downward force on the mass. As the mass falls, the potential energy it loses is converted into electrical energy in the generator. Since this change in potential energy depends on the mass, we observe that the 200-g mass does not light the lamp, the 500-g mass lights it only dimly, and the 1-kg mass lights it brightly. Once the circuit is broken and no current can flow, however, this energy no longer goes into producing electrical current, but is converted to kinetic energy of the mass, and the mass greatly accelerates. As noted above, Lenz’s law also helps to explain this behavior. As the mass falls and turns the wheel, the changing flux through the armature coils of the generator induces an EMF in them, which causes current to flow so that the magnetic field it produces opposes this changing flux. The magnetic field around the armature coils thus repels the magnetic field of the permanent magnets, and (through the tension in the string) opposes the force exerted by gravity on the mass. If the circuit is broken, no current can flow. There is thus no magnetic field around the armature coils, and no repulsion between them and the permanent magnets. Except for friction and rotational inertia in the generator and gear box, there is nothing to oppose the force of gravity on the mass, and it accelerates downward.

Ideally, this generator should produce a quantity of electrical energy equivalent to the change in gravitational potential energy of the mass over the distance through which it falls. This quantity is reduced, of course, by however much the efficiency of the generator is less than 100%. With the 1-kg mass at the end of the string, the generator produces a current of about 140 mA through the lamp at about 5.3 V, which corresponds to a power of about 0.74 watts. If the mass is released so that it falls the entire length of the string, it takes approximately 8.8 seconds to do so. This corresponds to a total energy output of (0.74 W)(8.8 s) = 6.5 J. The length of the string is six feet, or about 183 cm. The change in potential energy of the 1-kg mass during its descent is thus (1.0 kg)(9.8 m/s²)(0.183 m) = 18 J. This corresponds to an efficiency of about 36%. This is probably reasonable, since it is quite likely that the lamp and generator are not matched, and we are not operating the generator at its maximum efficiency.