Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
essentially the same as iÿ moves about. If we wish to study the motion of such a
body, and agree to ignore the motion of is center oŸ mass, there is only one thing
left for it to do, and that 1s to furn. We have to describe that. How? Suppose
there is some line in the body which stays put (perhaps it includes the center oŸ
mass and perhaps not), and the body is rotating about this particular line as
an axis. How do we defne the rotation? 'Phat is easy enough, for if we mark a
point somewhere on the obJect, anywhere except on the axis, we can always tell
exactly where the object is, if we only know where this point has gone to. The
only thing needed to describe the position of that point is an øngle. So rotation
consists of a study of the variations of the angle with time.
In order to study rotation, we observe the angle through which a body has
turned. Of course, we are not referring to any particular angle #ns¿de the object
itself; ¡% is not that we draw some angle øn the object. We are talking about the
angular change oƒ the position of the whole thing, from one tỉme to another.
Jirst, let us study the kinematics of rotations. 'Phe angle will change with
time, and just as we talked about position and velocity in one đimension, we
may talk about angular position and angular velocity in plane rotation. In fact,
there is a very interesting relationship between rotation in two dimensions and
one-dimensional displacement, in which almost every quantity has its analog.
First, we have the angle Ø which defnes how far the body has gone around; this
replaces the distance +, which defnes how far ¡it has gone ølong. In the same
manner, we have a velocity oŸ turning, œ = đØ/di, which tells us how mụuch the
angle changes in a second, just as ø = đs/d£ describes how fast a thing moves,
or how far it moves in a second. I the angle is measured in radians, then the
angular velocity œ will be so and so many radians per second. “The greater the
--- Trang 337 ---
angular velocity, the faster the object is turning, the faster the angle changes.
W© can go on: we can diferentiate the angular velocity with respect to time, and
we can call œ = dư /dt = d20/df2 the angular acceleration. That would be the
analog of the ordinary acceleration.
Q—Y C\|Vx ộ
x$P(x, y)
Að đã y
Fig. 18-1. Kinematics of two-dimensional rotation.
Now oŸ course we shall have to relate the dynamics oŸ rotation to the laws oŸ
dynamies of the particles of which the obJect is made, so we must fnd out how
a particular particle moves when the angular velocity is such and such. To do
this, let us take a certain particle which is located at a distance z from the axis
and say ÍÊ is in a certain location ?{z, ø) at a given instant, in the usual manner
(Fig. 18-1). If at a moment Af later the angle of the whole object has turned
through A0, then this particle is carried with it. It is at the same radius away
from ) as it was before, but is carried to Q. The first thíng we would like to know
is how much the distance + changes and how much the distance changes. lf @?P
is called r, then the length P@) is z A0, because of the way angles are defined.
The change in z, then, is simply the projection of z A0 in the z-direction:
Az = —PQsinØ = —r A0 - (0/r) =—ụA0. (18.6)
Similarly,
Au = +z A0. (18.7)
TÍ the object is turning with a given angular velocity œ, we fñnd, by dividing both
sides of (18.6) and (18.7) by A¿, that the velocity oŸ the particle is
Uy —= — and Uy = +u#. (18.8)
Of course if we want to fnd the magnitude of the velocity, we Just write
0= (J0ệ + 02 = V202 + 1333 = 0V #2 + J2 = ê†. (18.9)
--- Trang 338 ---
Tt should not be mysterious that the value of the magnitude of this velocity is (07;
in fact, it should be self-evident, because the distance that it moves is z.AØ and
the distance it moves per second is r A0/Af, or rứ.
Let us now move on to consider the dựngmics of rotation. Here a new concept,
ƒorce, must be introduced. Let us inguire whether we can invent something which
we shall call the #orque (L. torquere, to twist) which bears the same relationship to
rotation as force does to linear movement. Á force is the thing that is needed to
make linear motion, and the thing that makes something rotate is a “rotary force”
or a “twisting force,” i.e., a torque. Qualitatively, a torque is a “twist”; what is a
torque quantitatively? We shall get to the theory of torques quantitatively by
studying the øork done in turning an object, for one very nice way of delning a
force is to say how much work it does when it acts through a given displacement.
W© are going to try to maintain the analogy between linear and angular quantities
by equating the work that we do when we turn something a little bit when there
are forces acting on it, to the £orgue tỉmes the øngle it turns through. In other
words, the deflnition of the torque is goïng to be so arranged that the theorem oŸ
work has an absolute analog: force times distance is work, and torque times angle
1s goïng to be work. That tells us what torque is. Consider, for instance, a rigid
body of some kind with various forces acting on it, and an axis about which the
body rotates. Let us at frst concentrate on one force and suppose that this force
is applied at a certain point (z,). How much work would be done iŸ we were to
turn the object through a very small angle? 'Phat is easy. he work done is
AW = F„ Az + lụ A. (18.10)
W©e need only to substitute Bqs. (18.6) and (18.7) for Az and A¿ to obtain
AW = (zty— uEF„)A0. (18.11)
That is, the amount of work that we have done is, in fact, equal to the angle
through which we have turned the object, multiplied by a strange-looking combi-
nation of the force and the distanece. 'Phis “strange combination” is what we call
the torque. So, defining the change in work as the torque times the angle, we
now have the formula for torque in terms oŸ the forces. (Obviously, torque is not
a completely new idea independent of Newtonian mechanics—torque must have
a defnite deflnition in terms of the force.)
'When there are several forces acting, the work that is done 1s, of course, the
sum of the works done by all the forces, so that AW will be a whole lot of terms,
--- Trang 339 ---
all added together, for all the forces, cach oƒ uhách is proportional, houeuer,
to A0. W© can take the AØ outside and therefore can say that the change in the
work is equal to the sum of all the torques due to all the diferent forces that are
acting, times A0. 'This sum we might call the total torque, 7. Thus torques add
by the ordinary laws of algebra, but we shall later see that this is only because
we are working in a plane. lt is like one-dimensional kinematics, where the forces
simply add algebraically, but only because they are all in the same direction. lt
1s more complicated in three dimensions. 'Phus, for two-dimensional rotation,
T=À T¡. (18.13)
lt must be emphasized that the torque is about a given axis. lfa different axis is
chosen, so that all the ø; and ¿ are changed, the value of the torque is (usually)