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5.4Transmission of Forces by Low-Mass Objects
You’re walking your dog. The dog wants to go faster than you do, and
the leash is taut. Does Newton’s third law guarantee that your force on your
end of the leash is equal and opposite to the dog’s force on its end. If they’re
not exactly equal, is there any reason why they should be approximately
equal.
If there was no leash between you, and you were in direct contact with
the dog, then Newton’s third law would apply, but Newton’s third law
cannot relate your force on the leash to the dog’s force on the leash, because
that would involve three separate objects. Newton’s third law only says that
your force on the leash is equal and opposite to the leash’s force on you,
F
yL
= – F
Ly
,
and that the dog’s force on the leash is equal and opposite to its force on the
dog
F
dL
= – F
Ld
.
Still, we have a strong intuitive expectation that whatever force we make
on our end of the leash is transmitted to the dog, and vice-versa. We can
analyze the situation by concentrating on the forces that act on the leash,
F
dL
and F
yL
. According to Newton’s second law, these relate to the leash’s
mass and acceleration:
F
dL
+ F
yL
= m
L
a
L
.
The leash is far less massive then any of the other objects involved, and
if m
L
is very small, then apparently the total force on the leash is also very
small, F
dL
+ F
yL
˜
0, and therefore
F
dL
˜
– F
yL
.
Thus even though Newton’s third law does not apply directly to these
two forces, we can approximate the low-mass leash as if it was not interven-
ing between you and the dog. It’s at least approximately as if you and the
dog were acting directly on each other, in which case Newton’s third law
would have applied.
In general, low-mass objects can be treated approximately as if they
simply transmitted forces from one object to another. This can be true for
strings, ropes, and cords, and also for rigid objects such as rods and sticks.
If you look at a piece of string under a magnifying glass as you pull on
the ends more and more strongly, you will see the fibers straightening and
becoming taut. Different parts of the string are apparently exerting forces
If we imagine dividing a taut rope up into
small segments, then any segment has
forces pulling outward on it at each end.
If the rope is of negligible mass, then all
the forces equal +T or -T, where T, the
tension, is a single number.
Section 5.4Transmission of Forces by Low-Mass Objects