BALANCING, AND SOME QUESTIONS OF POTENTIAL ENERGY—HILL-CLIMBING.
It seems pertinent at this point to make some further distinction
between two distinctive classes of road wheels. The conception in the
mind of man of road carriages which require an element of balancing
was a recent event in the development of vehicles in general, and the
similarity of the words bicycle and tricycle, together with the fact
that both are included in the generic term velocipede, has led many
to overlook a distinction of balancing, which should class them under
very different heads. Both are velocipedes if we mean machines run by
foot-power; both are man-motors in the light that human force or energy
actuates them; but the two-wheel single-track machine must employ a
particular faculty on the part of the rider, not required in running
one of stable equilibrium.
It seems superfluous at this stage of development of the art to
enlarge upon the fact that a bicycle has to be balanced by a particular
action not required in any other form of carriage; but when inventors
will keep on getting up means to lock the steering device, and riders
will persist in reminding us that the steering head “moves too easily,”
it is severely pertinent to remark that while a certain law of whirling
bodies might show us that a wheel will not fall over quite so quickly
when rolling as when standing still, yet it is not this law so much as
the action of steering that differentiates the bicycle, or single-track
carriage, from other machines. The action of the handle-bar while in
motion does substantially, in balancing the bicycle, what you would do
if you were balancing a cane vertically on the end of your nose: if the
cane starts to fall, you run in that direction with your nose till you
get under the centre of gravity again. But the bicycle can only fall
sideways, so, when it tends to fall in that way, or when the centre of
gravity gets to one side of the vertical line from the point of support
on the ground, you cannot run directly sideways with the support as you
would in the cane illustration, but you can run indirectly sideways,
nevertheless, with the point of support, the only difference being that
you must run considerably forward at the same time in order to shift
the lower extremity, or point of contact and support, in that direction.
After considerable discussion of this apparently simple subject with
eminent gentlemen well qualified to speak on such topics, the following
appeals to my mind as a more definite and complete explanation than
that given in the nose and cane case, bringing in an element of the
problem omitted above, to wit: in running the point of support of
contact across and under, as it approaches the vertical plane of
gravity and general forward momentum, the steering wheel lies slightly
across this plane, and its own plane is still out of vertical, leaning
a little, as it did before, with the centre of gravity back of the
point of support; the forward momentum then throws the entire system
upright. In rapid running this momentum does a large proportion of the
work, and it has been vigorously maintained that all balancing is due
to this element; for small motions, however, the cane explanation is
quite sufficient.
The foregoing explanation of uprighting the bicycle is, to my mind,
almost _entirely_ independent of any law of whirling bodies as
generally understood.
An article showing that this subject is not devoid of interest or
obsolete is given below from the _Bicycling World_, in which I think
the law of whirling bodies will apply. “The Rochester wheelmen debated
the question, ‘Why does a bicycle stand up while rolling and fall down
as soon as onward motion ceases?’ The answer decided to be correct
was, that ‘the bottom of the wheel can have no side motion because it
rests on the ground; and since the bottom is constantly becoming the
top and the top the bottom, if the upper part of the wheel gets any
lateral motion, it is checked by being brought round upon the ground
again before the motion has too much influence.’” I do not suppose
this ingenious decision, rendered by the high and mighty Solons of the
Rochester Club, was a serious one; however, we do find that just such
logic is quite common.
It is not plain whether the question discussed was that of a bicycle
with or without a man upon it, but I take it to be the latter. Some of
the gentlemen had no doubt noticed that to give the machine a shove
it would keep upright for a longer time running than when standing
unsupported. This is purely a case of the law that whirling things
tend to keep their own plane, as illustrated in the gyroscope and
the spinning top. In the running bicycle without a man upon it to
constantly rectify its position, the principle is simply one of the
parallelogram of rotations. If the wheel from any external force starts
to fall over, or, in other words, to revolve around a horizontal
line normal to its geometric axis, then, since the wheel is already
revolving about its axis in the axle, the resultant of these two
rotations will be a rotation about an axis inclined to the former axis
of the wheel, which means that the wheel will begin to circle around
a centre at some distance from the wheel on the side towards which
it starts to fall. This new axis about which the wheel revolves will
of course be in a plane perpendicular to the new plane of the wheel,
and will be inclined downward from the horizontal plane through its
centre, so that the wheel is no longer running in a vertical plane. The
rotation about the centre outside of the wheel, towards which centre
the wheel leans, brings into play a centrifugal force acting to upright
the wheel; that is, to bring it back to a vertical plane. Now, if the
wheel be run along a straight groove, so that circling around a centre
is prevented, then it will fall as quickly as when standing still; or
if, in the bicycle, the steering-wheel be locked so that it will not
turn out of the plane of the two wheels, there would be no uprighting
resultant, and the machine, according to Newton’s law of independent
forces, would fall.
SOME QUESTIONS OF POTENTIAL ENERGY, MOMENTUM, AND HILL-CLIMBING.
When a cyclist climbs a hill, he not only overcomes the friction which
would be generated if he travelled over the same length of level road
surface, but he ought to be supposed to establish a certain amount
of potential energy, or energy against gravity, and therefore should
lose none. Yet he does lose considerable somewhere or he would not
dread the hilly road as he does. In this matter of potential energy in
hill-climbing upon a cycle, the subject assumes a different aspect from
that of rolling on or off obstructions, as in rough-road riding treated
of elsewhere. In climbing a hill there is no loss of momentum from a
too sudden change in its direction; the matter of inertia does not
figure in the case in any way, and we have a mere question of the rise
and fall of a weight under certain modifications, said weight being the
rider and his machine, said rise the ascent of the hill, and the fall
the descent thereof. In a purely physical sense, then, we store up a
certain amount of energy, or, in other words, put so much energy to our
credit as against gravity, and theoretically we have a right to expect
to get the benefit of it.
To illustrate this potential energy, suppose we place a pulley at
the top of a hill and a rider at each end of a rope running over the
pulley, with one man at the bottom starting up and the other at the
top starting down the same hill. The descent of one man would draw
the other up, excepting that each would have to work only just enough
to make up the loss from friction, as he would in case the road were
level and of equal length. I have little doubt that in such a pulley
arrangement there would be much less loss of power and energy than
riders now experience in the actual practice of hill-climbing. To
illustrate with one man how the potential energy should be returned and
thereby benefit the rider, let us place him at the top of a hill at the
bottom of which another hill of the same height begins, whence, by the
acceleration of gravity, the rider ought to find himself at the bottom
of the first hill with an amount of momentum acquired that would send
him to the top of the next; in other words, we might naturally expect
when we roll down one incline to roll just as far up another of the
same grade, or of the same vertical height regardless of the grade, or
else we should expect a return of the energy in sending us capering
over a level road without further labor, until the kinetic energy is
exhausted. We find, however, that such a desirable result does not
appear, and we notice that, however long, beyond a certain limit, the
hill may be, we have no more momentum or kinetic energy at our disposal
than we would in the case of a shorter hill. To what can this loss be
attributed? There is but one visible cause,—to wit, our work against
the air.
If all riding were done in a vacuum, we would more nearly get back our
energy, but somehow or other the vacuum is generally in the rider and
doesn’t count, so there is an end to that. The rider, then, loses the
momentum he would acquire from gravity because the friction of the air
is resisting his progress at the rate of, or according to, the square
of his velocity. In order to store up all the energy in a falling body
we must allow gravity to increase the velocity as the square root of
the distance. But it is easily seen that a rate of speed will soon be
reached such that the air by impact will entirely annul all increase
of velocity, and therefore all of the momentum we can expect to have
at the bottom of the hill is just that which was acquired at the time
and point at which the impact of the air balanced the accelerating
force of gravity. This will soon come to pass, even omitting other
friction, which, in connection with hill-climbing, we can afford
to omit with good reason, because we should expect to have that to
overcome if the road were level. The mere difference in the length of
the surface travelled over will not bother a cyclist if it be a good
level road, so we must blame it all on the air; I see no other way
out of it. No manner of springs or anti-vibrators will help us out of
this difficulty. If our rider puts on the brake, then of course there
is no question as to where the work goes; but, as we all know, with a
safe machine and an expert rider this is not often done in an ordinary
country.
In defence of our theory of loss of energy on very long hills, observe
the fact that a mere rolling road is not generally despised by the
cyclist; in fact, many prefer it to a dead level, the writer being
decidedly one of their number. The short intervals of labor and rest,
the continual barter and sale with gravity, in the transfer of energy
to and fro, is not by any means an uncomfortable diversion to either
our minds or bodies; but when we come to suffer the usurious interest
demanded by the action of the air against us, we simply draw the line,
and go by another road, even though the surface thereof be not of the
most inviting character.
Some ingenious mechanics have devised mechanism whereby they propose
to store up the power lost in the brake action; but it is doubtful if
any riders would care for it after they become expert and daring, which
they all do in course of time in spite of all admonition against undue
risk.
Speaking of potential energy and momentum, we naturally come upon the
question of machine weight. It is a peculiar fact that the weight of
the man does not form so important a part in the bicycle exercise as
that of the machine, so that if a rider be heavier by twenty pounds
than another, it will not generally count against him; but if that
weight is in the machine, competition is out of the question. Nature
seems to make up in muscle, or supply of energy in some way, for the
extra weight in the man, but said nature is not so clever when this
weight is outside of him.
It is sometimes thought that a heavy man or a heavy machine will
descend a hill faster than a lighter. This is not reasonable. The
accelerating force of gravity being independent of the mass, the
heavy system will have the same velocity at the bottom, and momentum
being represented by mass, times velocity, the increased mass will
increase the momentum; but the speed is the same: this extra momentum
is required in raising the heavier system to the same height as the
lighter. But even if the rider should get the benefit of all the energy
he stores in climbing a hill, there is still an indisputable objection
to a heavy wheel,—to wit, a man can labor long and continuously at a
strain within reasonable limits, and can do a large amount of work
thereby; but to strain the system beyond those limits, and attempt
to store up too much energy in too short a space of time, is to make
nature revolt, resist the imposition, and refuse to be appeased for
some time to come and often not at all; in short, an overstrain is bad,
and by a heavy machine, no matter what amount of energy you may store
up at the top of the hill, if in so doing nature has been overtaxed,
it will result disastrously. So we see that, outside of all mechanical
questions of momentum and potential energy, there is a vital objection
to heavy machines on purely physiological grounds.