Omaha daily bee. (Omaha [Neb.]) 187?-1922, 1915-03-09
Curiosity of Motion Illustrated
How the Top of Wheel Moves Twice as Fast
as the Hub and Stops for an Instant at the Bottom

thirty miles an hour by virtue OF its TO.
rational motion about its axle. making a
! sum OF sixty or, to put It another way.
while tho point is moving thirty miles an
| hour i'm forward direction with regard
| e. the axle, the axle itself is moving
thirty miles an hour I'm the same direc-
| ton.
I This coincidence OF the two velocities
last only for an infintcsimal instant. be-
cause. after passing .he top, the point
must curve downward and then less and
less OF its aNial. or rotational. motion
will coincide with tho forward motion
of translation, until. when Il touches the
bottom, the two velocities. equal in
magnitude, will be exactly opposed I'm
direction. Their algebraic sum is then
zero, and for an inappreciable instant the
point will be at rest with reference to the
ground.
| To show how the speed OF the rim OF
a wheel may be calculated at any point
around the circumference take In your
diagram, the point which lies directly
ahead OF the center Of the wheel. Here
/ tho rotauonal velocity il, for the in
slant. directed straight downward. while
the translatory velocity remains uni
q
TO one another and they are equal IN
magnitude. Hence they may be regarded
as representing two adjacent sides of a
/ square, each side being equal to thirty
units Now. by the principle OF the COM
position OF motions, the resultant Of the
combination OF the rotational and trans
latory velocities Will DC represented, I'm
magnitude and direction. by the diagonal
OF the square
To calculate the length OF the diagonal
we have only to add the squares OF the
two adjacent sides and extract the square
root OF the sum. The square of each side
| is 900, the sum Of the squares is t.SO0 the
square root OF 800 is ~ 4265. which
represents the speed, in miles per hour,
OF the point on the rim. when It is in
the position shown a similar method will
show the speed and direction OF the re-
sultant motion at any place on the rim.
suppose that It is the front wheel OF
a carriage with which we are dealing
then how fast win the tire of the wheel
be passing the elbow of the driver? You
may be tempted by an inspection of the
diagram to reply, sixty miles an hour"
but that would be wrong. The true an
swer Is thirty miles an hour. because
the driver shares the general motion of
translation, and the rim of the wheel
passes his elbow with only its velocity
of rotation.
Tho motion IL space, with reference to
the ground, of a point on the circumfer-
ence OF n running wheel Is very cur
ious subject of mathematical study. The
combination of the two motions, which
we have been considering, causes suca a
point To describe very peculier curve,
called a oycloid. Starting with the point
in contact with the ground, the cycloid
rises, with gradually decreasing curva-
ture. until the point Is at the top of the
wheel. Immediately Il begins TO curve
down again until It once more touches
, the ground Every point in the rim of the
wheel Is continually moving in a
cycloidal curve. and the directions and
velocites which we have calculated in
the case supposed are those pertaining
to oycloid at the point designated,
Among its other peculiar properties tho
, cycloid is the curve OF quickest descent
This means that II two points are sltu-
ated one above the other, without being
p in n vertical line, a heavy body will
descend from the upper to the lower
, point along an arc of an inverted cycloid
quicker than along any other curve, and
quicker even than along a straight line.