Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
Like torque, angular momentum depends upon the position of the axis about
which it is to be calculated.
Before proceeding to a treatment of more than one particle, let us apply the
above results to a planet going around the sun. In which direction is the force?
'The force is toward the sun. What, then, is the torque on the object? Of course,
this depends upon where we take the axis, but we get a very simple result if
we take i% at the sun itself, for the torque is the force times the lever arm, or
the component of force perpendicular to z, times z. But there is no tangential
force, so there is no torque about an axis at the sun! "Therefore, the angular
momentum of the planet goïng around the sun must remain constant. Let us see
what that means. The tangential ecomponent of velocity, times the mass, times
the radius, will be constant, because that is the angular momentum, and the
rate of change of the angular momentum ¡is the torque, and, in this problem,
the torque is zero. OÝ course since the mass is also a constant, this means that
the tangential velocity times the radius is a constant. But this is something we
already knew for the motion of a planet. Suppose we consider a smaill amount of
--- Trang 343 ---
time Af. How far will the planet move when it moves from ? to @ (Fig. 18-3)?
How mụuch ørea will it sweep through? Disregarding the very tiny area QQ“P
compared with the mụch larger area @P@), it is simply half the base P€) times
the height, Of. In other words, the area that is swept through in unit time will
be cqual to the velocity times the lever arm of the velocity (times one-half). 'Thus
the rate of change of area is proportional to the angular momentum, which 1s
constant. So Kepler”s law about equal areas in equal times is a word description
of the statement of the law of conservation of angular momentum, when there is
no torque produced by the force.
18-4 Conservation of angular momentum
Now we shall go on to consider what happens when there is a large number
of particles, when an object is made of many pieces with many Íorces acting
between them and on them from the outside. Of course, we already know that,
about any given fxed axis, the torque on the ¿th particle (which is the force on
the ¿th particle times the lever arm of that force) is equal to the rate oŸ change of
the angular momentum of that particle, and that the angular momentum of the
¿th particle is 10s momentum times its momentum lever arm. NÑow suppose we
add the torques 7; for all the particles and call ít the total torque 7. Then this will
be the rate of change of the sum of the angular momenta of all the particles L,
and that defnes a new quantity which we call the total angular momentum Ù.
Just as the total mornentum of an object is the sum of the momenta of all the
parts, so the angular momentum is the sum of the angular momenta of all the
parts. hen the rate of change of the total Ù is the total torque:
dLị dù
T=Ồ 7=” . (18.18)
Now it might seem that the total torque is a complicated thing. 'There are all
those internal forces and all the outside forces to be considered. But, if we
take Newton”s law of action and reaction to say, not simply that the action
and reaction are equal, but also that they are đirccted exactlU oppositelu along
the seme line (NÑewton may or may not actually have said this, but he tacitly
assumed it), then the Ewo £orgues on the reacting objects, due to their mutual
interaction, will be equal and opposite because the lever arms for any axis are
equal. Therefore the internal torques balance out pair by pair, and so we have
the remarkable theorem that the ra#e oƒ chưnge oƒ the totaÌ œngular mmomentum
--- Trang 344 ---
about am a#is is cqual to the external torque about that azis!
T= `T¡ = To„y = dL/dị. (18.19)
Thus we have a very powerful theorem concerning the motion of large collections
of particles, which permits us to study the over-all motion without having to
look at the detailed machinery inside. This theorem is true for any collection of
objects, whether they form a rigid body or not.
One extremely important case of the above theorem is the la+ oƒ conseruation
öƒ angular mmomentum: 1ƒ no external torques act upon a system of particles, the
angular momentum remains constant.
A special case of great importance is that of a rigid body, that is, an object
of a defñnite shape that is just turning around. Consider an object that is ñxed
in its geometrical dimensions, and which is rotating about a ñxed axis. Various
parts of the object bear the same relationship to one another at all times. Ñow
let us try to ñnd the total angular momentum of this object. If the mass of one
OÝ its particles is rn¿, and its position or location is at (#;, ¡), then the problem
1s to fnd the angular momentum of that particle, because the total angular
tmmomentum is the sum of the angular momenta of all such particles in the body.
For an object going around in a circle, the angular momentum, of course, is the
mass times the velocity times the distance from the axis, and the velocity is equal
to the angular velocity times the distanece from the axis:
L¡ = tmju¡r¿ = mạrỆằ), (18.20)
or, sunming over all the particles ?, we get
L= Tu, (18.21)
T=Ồ mịr?. (18.22)
This is the analog of the law that the momentum is mass times velocity.
Velocity is replaced by angular velocity, and we see that the mass is replaced
by a new thing which we call the rmornen‡ oƒ inertia T, which is analogous to
the mass. Equations (18.21) and (18.22) say that a body has inertia for turning
which depends, not just on the masses, but on hou ƒar a+UdU the are from the
axis. So, IÝ we have two objects of the same mass, when we put the masses
--- Trang 345 ---
lÝ 5) sĩ
Fig. 18-4. The “inertia for turning” depends upon the lever arm of the
masses.
farther away from the axis, the inertia for turning will be higher. 'This is easily
demonstrated by the apparatus shown in Fig. 18-4, where a weight Mƒ is kept
from falling very fast because it has to turn the large weighted rod. At first, the
masses ?w are close to the axis, and M⁄ speeds up at a certain rate. But when
we change the moment of inertia by putting the wo masses rn much farther
away from the axis, then we see that MỸ accelerates much less rapidly than it did
before, because the body has much more inertia against turning. The moment of
inertia is the inertia against turning, and is the sum of the contributions of all
the masses, times their distances sguared, from the axis.
'There is one important diference between mass and moment of inertia which
is very dramatic. The mass of an object never changes, but its moment of inertia,
can be changed. lf we stand on a frictionless rotatable stand with our arms
outstretched, and hold some weights in our hands as we rotate slowly, we may
change our moment of inertia by drawing our arms in, but our mass does not