pin, Fig. 140. Since the velocity of every point of the rod in the direction of its length is the same, the projections of v and v1 on the

rod are equal. The rela

tions will not be altered if the figure v1ED be turned through 90° and E be placed at O and V1 be made to coincide with OF and drawn to such a scale as to equal

r. Then v will fall on

A

ย

FIG. 141

OG and will be of a length equal to that cut off on OG by AF produced (see Fig. 141).

Then

υ

V1

sin (α + $) cos o

v = v1 (sin α + cos a tan 6).

For small values of we may replace tan

But

sin & r sin α

=

by sin o, so that

For a = 30°, v = - 30.5 ft. per second, and a =

1306 ft. per (second)2.

The three equations in Σx, Σy, and Σmom, now give N' = 9660 lb., N1 = 58,137 lb., and T = the resultant pressure on the crank pin to be 60,900 lb. - 18,151 lb. Compounding N1 and T, we get

=

The

negative signs for N1 and T indicate that the arrows were assumed in the wrong direction.

α

Problem 182. Show that the values of a that make o a maximum or minimum, when the motion of the crank is assumed constant, are π, 3π, etc., and o, 2 π, etc.

Problem 183. Find what values of a will make a maximum or minimum. Locate the crosshead for these values.

Problem 184. Find values for T, N', N1, and the resultant pressure on the crank-pin when α = π and when α = 0. Use the above data. Assume a force of friction F acting on the cross.06 N'. In the above case when α = 30°, what is N1, and T?

Problem 185. head, such that F

the value of F, N',

Problem 186.

=

Suppose the steam pressure zero, find T, N', Ní,

if is the same.

and the resultant crank-pin pressure, @1

be

120. Body Rotating about an Axis - One Point Fixed. — We shall now consider the case of a body rotating about an axis when only one point of that axis is fixed. Consider the equations 1, 2, 3, 4, 5, and 6 (Art. 104) and recall that a body acted upon by any system of forces may considered as being acted upon by a single force and a single couple (Art. 36). If one point of the axis is fixed, the single force will act at this point, but the effect of the single couple will be to move the axis of rotation about this point. The components of the moment of the couple are shown in the right-hand side of equations (4) and (5). If these reduce to zero, the couple vanishes and rotation continues about the original axis of rotation even though only one point of that axis is fixed. But this can happen only

=

when Sazd M O and also SyzdM = 0. We may say, then, that a body rotating about an axis one point of which is fixed, when no forces are acting to produce rotation, will I continue to rotate about that axis with a constant angular velocity w.

121. Gyroscope. — The gyroscope illustrated in Fig. 142 consists of a metal wheel A mounted on an axis BB, fixed

at one point to the stand C. The weight D serves to balance the wheel about the point of support. The wheel A is very delicately mounted, so that there is little friction. It is set in motion by means of a. cord wound around its axle, as in the case of an ordinary top.

B

D

E

B

FIG. 142

When the weight D exactly balances A so that BB is horizontal, we have a case of a body rotating about an axis fixed at one point, with no external forces acting. According to the previous article, the body continues to

rotate about that axis.

If, however, the weight D does not balance A, so that BB is not horizontal, the axis of rotation changes, since in that case the force of gravity tends to turn BB about a horizontal axis through E perpendicular to BB. As a result of the two rotations, the body tends to turn about a new axis, so that BB turns about the point E horizon

tally. All this is in accordance with Art. 95, where we saw that the resultant of two angular velocities was an angular velocity given by the diagonal of a parallelogram constructed upon the two velocity arrows as sides. This rotation of the gyroscope about the vertical axis through E is known as precession. The student may reproduce the above results experimentally by taking a bicycle wheel mounted upon its axle. Suspend one end of the axle by a string and hold the other in the hand, so that the axle is horizontal. With the other hand now give the wheel a spin. If the axle remains horizontal, the wheel contin ues to spin about the same axis, but if the hand supporting one end of the axle be removed, the wheel continues to rotate about its own axis while the axis rotates about the suspending string. In other words, the wheel has a motion of precession.

The motion of the bicycle wheel is explained in the same manner as that used in explaining the precession of the gyroscope.

122. The Spinning Top. The student will be interested in seeing that the motion of a spinning top, with which all are familiar, is also capable of the same explanation. Let the top be represented, as in Fig. 143, with its point at O, and suppose it has an angular velocity w about its own axis. If it is rotating sufficiently rapidly, and its axis is vertical, it "sleeps," or continues to revolve about that same axis. If, however, the axis be tilted slightly from the vertical, the weight of the top G and the reaction of the floor constitute an unbalanced couple tending to make it revolve about an axis through O perpendicular to the

paper.

The result of these two rotations is to cause the top to always tend to revolve about an axis a little in front of the geometrical

Ꮓ

W1

well known that when the top has been so disturbed,

-X

if spinning rapidly, it

FIG. 143

moves about OZ very slowly, and gradually takes the vertical position again.

When the velocity of rotation w finally becomes small, the irregularities of the support throw the axis out of the vertical, and the action of the unbalanced couple causes precession. At first the precession is very slow, but gradually increases as a decreases until the top falls.

123. Motion of Earth. A brief presentation of this subject would be incomplete without mentioning the precession of the earth's axis. In Fig. 144 let S represent the sun and E the earth, with its axis slightly inclined to the vertical. The earth is a spheroid with its axis of rotation as its short axis. Consider the ring of matter near the equator which if cut off would make the earth spherical. The attraction of the sun for this ring of matter is greater on the side nearest the sun. This causes the earth to be acted upon by an unbalanced force F, and tends to cause a rotation of the earth about an axis through O perpendicular to the paper. As a result the axis of rotation is moved