Chapter 20
Torque & Circular Motion
124
20
Torque & Circular Motion
The mistake that crops up in the application of Newton’s 2
nd
Law for Rotational
Motion involves the replacement of the sum of the torques about some particular
axis,
∑
τ
, with a sum of terms that are not all torques.
Oftentimes, the errant
sum will include forces with no moment arms (a force times a moment arm is a
torque, but a force by itself is not a torque) and in other cases the errant sum will
include a term consisting of a torque times a moment arm (a torque is already a
torque, multiplying it by a moment arm yields something that is not a torque).
Folks that are in the habit of checking units will catch their mistake as soon as
they plug in values with units and evaluate.
We have studied the motion of spinning objects without any discussion of torque.
It is time to
address the link between torque and rotational motion.
First, let’s review the link between force
and translational motion.
(Translational motion has to do with the motion of a particle through
space. This is the ordinary motion that you’ve worked with quite a bit.
Until we started talking
about rotational motion we called translational motion “motion.”
Now, to distinguish it from
rotational motion, we call it translational motion.)
The real answer to the question of what
causes motion to persist, is nothing—a moving particle with no force on it keeps on moving at
constant velocity.
However, whenever the velocity of the particle is changing, there is a force.
The direct link between force and motion is a relation between force and acceleration.
The
relation is known as Newton’s 2
nd
Law of Motion which we have written as equation 141:
∑
=
F
harpoonrightnosp
harpoonrightnosp
m
1
a
in which,
a
harpoonrightnosp
is the acceleration of the object, how fast and which way its velocity is changing
m
is the mass, a.k.a. inertia, of the object.
m
1
can be viewed as a sluggishness factor, the
bigger the mass
m
, the smaller the value of
m
1
and hence the smaller the acceleration of the
object for a given net force.
(“Net” in this context just means “total”.)
∑
F
harpoonrightnosp
is the vector sum of the forces acting on the object, the net force.
o
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Chapter 20
Torque & Circular Motion
125
We find a completely analogous situation in the case of rotational motion.
The link in the case
of rotational motion is between the angular acceleration of a rigid body and the torque being
exerted on that rigid body.
∑
=
τ
harpoonrightnosp
harpoonrightnosp
I
1
a
(201)
in which,
a
harpoonrightnosp
is the angular acceleration of the rigid body, how fast and which way the angular velocity is
changing
I
is the moment of inertia, a.k.a. the rotational inertia (but not just plain old inertia, which is
mass) of the rigid body.
It is the rigid body’s inherent resistance to a change in how fast it
(the rigid body) is spinning.
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 Fall '08
 RABE
 Physics, Angular Momentum, Circular Motion, Moment Of Inertia, Rotation, Angular velocity

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