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Rotational Motion
In this chapter the motion is constrained to move in a circular path. It is very much
like the chapter on linear motion in that the object can only move in one of two
directions, clockwise or counterclockwise. We will form motion descriptors like we did
for linear motion.
Motion Descriptor
Distance (x)
Speed (v)
Acceleration (a)
Just as v 
Units
meters
m/s
m/s2



Rotational Descriptor
Angular Distance ()
Angular Velocity ()
Angular Acceleration ()
Units
rad
rad/s
rad/s2
x

v

, so  
. Similarly, a 
becomes  
.
t
t
t
t
Angular velocity and acceleration are vectors. To determine the direction of each,
curl the fingers of your right hand in the direction of rotation. Your right thumb then
points in the direction of  or .
Basic kinematical equations can be developed for pure rotation as a result of the
equivalences in the table above. So we get
   o  o t  1 2 t 2
  o  t
We learned that inertia is a property of matter that resists changes in motion and that
inertia is identical to mass. When the motion is pure rotation, the equivalent concept is
rotational inertia, the property of an object that resists changes in rotation. We measure
rotation inertia with the moment of inertia (I). I depends on the object’s mass and mass
distribution, and the axis of rotation. The easiest to compute is the mass on a string.
Make the axis of rotation a distance r away from the mass and the moment of inertia is I
= mr2. Other shapes and rotational axes require the use of calculus, but all take the form I
= f(mr2), where f is a fraction less than one. Remember the basic definition of inertia
here and many situations become clear.


Torque is the product of force ( F ) and lever arm ( d ).

 
  F d
Notice that torque is also a vector. The operation is often called the vector (or cross)
product. The general rules for finding the lever arm are:
1. Draw in the line of action of the force. This is a line through the force extending
in both directions.
2. Locate the axis of rotation (pivot point).
3. The lever arm is the shortest distance from the axis to the line of action.
Lever Arm
Unbalanced torques produce angular accelerations just as unbalanced forces produce
linear accelerations. Newton’s Second Law for the pure rotational case looks like




  I instead of F  ma .
Newton showed that all of the mass of an object acts as if it were concentrated in a
single point called the center of mass. A useful way of finding the center of mass of an
object is to take advantage of the fact that gravity can only point vertically. Suspend the
object by a point on the edge. If the center of mass is not along the vertical line through
the point of support, then a torque exists and the object rotates. When the center of mass
does fall along this line (the line of action for the body’s weight), the lever arm drops to
zero and no torque exists. Repeat this test for one or more other suspension points and
you can quickly find the center of mass.
An object is stable against rotations if the vertical line through the center of mass
strikes the base of support.
Centripetal force is any force that causes the object to move in a circle. The word
literally means “center seeking” and can be provided by many different forces or force
combinations. Using Newton’s Second Law, we can write Fc = mac = mv2/r = mr2.
“Centrifugal (center fleeing) force” is a fictitious force experienced by an object located
in an accelerated reference frame. As viewed from the outside of the system, these
“centrifugal forces” can be seen to be simple applications of Newton’s First Law. As an
example, while taking a turn in your car, you seem to be push away from the center of the
circle through which you are turning. But you are trying to move in a straight line as
dictated by the 1st Law and the outer car door is forcing you to turn. As it presses against
you, you also press against it, and it feels like a force.
Sometimes we can take advantage of the centrifugal force. In the future when we
build space habitats that rotate with constant , the occupants experience an outwardly
directed “force.” They would feel pressed against the outer rim of the habitat and would
experience the sensation of weight due to this force. The centripetal force is proportional
to r2 and would be greater the farther from the center the occupant lives. This simulated
gravity would, therefore, vary with radius in the habitat, being zero right at the center.


Angular momentum is the rotational equivalent to linear momentum. Since p  mv ,


we can write angular momentum as L  I . The direction of L is the same as the
direction of  given by the right hand rule. Recall that an alternate form of Newton’s 2nd
 p
Law is F 
and that from this we defined impulse as
t



 L
and thus
p  F  t . We now realize that in the purely rotational case  

t


rotational impulse is L    t . In systems where there are no external torques ( =
0), angular momentum must by conserved (L = 0). Look over the various examples of
angular momentum conservation to see this principle in action. For example, the inside
of the barrel of a gun has a heliacal groove cut into it called rifling. The purpose of the
rifling is to make the bullet rotate. Once the shell is ejected angular momentum is
conserved. The vector of angular momentum does not want to change, making the flight
truer. Without the rotation the shell would tend to tumble. The same arguments apply to
throwing a football.
Rotational physics can be approached from the point of view of substitution of
variables into the equivalent linear equations. Below is a table of such substitutions