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\Rotational Motion
Rotational Inertia and
Newton’s Second Law
 In linear motion, net force and mass determine
the acceleration of an object.
 For rotational motion, torque determines the
rotational acceleration.
 The rotational counterpart to mass is rotational
inertia or moment of inertia.
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

Just as mass represents the resistance to a change in linear
motion, rotational inertia is the resistance of an object to
change in its rotational motion.
Rotational inertia is related to the mass of the object.
It also depends on how the mass is distributed about the
axis of rotation.
Rotational Inertia and
Newton’s Second Law
 The resistance to a change in rotational motion
depends on:


the mass of the object;
the square of the distance of the mass from the axis of
rotation.
 For an object with its mass concentrated at a point:
 Rotational inertia = mass x square of distance from axis
I = mr2
 The total rotational inertia of an object like a merry-goround can be found by adding the contributions of all
the different parts of the object.
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Rotational Inertia and
Newton’s Second Law
 Newton’s second law for linear motion:
Fnet = ma
 Newton’s second law for rotational motion:
 The net torque acting on an object about a given
axis is equal to the rotational inertia of the object
about that axis times the rotational acceleration of
the object.
net = I
 The rotational acceleration produced is equal to the
torque divided by the rotational inertia.
Example:
a baton with a mass at both ends
 Most of the rotational
inertia comes from the
masses at the ends.
 A torque can be applied at
the center of the rod,
producing a rotational
acceleration and starting
the baton to rotate.
 If the masses were moved
toward the center, the
rotational inertia would
decrease and the baton
would be easier to rotate.
Conservation of Angular
Momentum
How do
spinning
skaters or
divers
change
their
rotational
velocities?
I = mr2
Angular Momentum
 Linear momentum is mass (inertia) times
linear velocity:
p = mv
 Angular momentum is rotational inertia times
rotational velocity:
L = I
 Angular momentum may also be called
rotational momentum.
 A bowling ball spinning slowly might have the
same angular momentum as a baseball
spinning much more rapidly, because of the
larger rotational inertia I of the bowling ball.
Conservation of Angular
Momentum
 Linear momentum is conserved if the net external
force acting on the system is zero.
 Angular momentum is conserved if the net external
torque acting on the system is zero.
Inertia m : Fnet  ma
Inertia I :  net  I
p  mv
L  I
If Fnet  0,
p  constant
If  net  0, L  constant
1 2
1 2
KE  mv
KE  I
2
2
Angular momentum is conserved by
changing the angular velocity
 When the masses are
brought in closer to the
student’s body, his
rotational velocity increases
to compensate for the
decrease in rotational
inertia.
 He spins faster when the
masses are held close to
his body, and he spins
more slowly when his arms
are outstretched.
Angular momentum is conserved by
changing the angular velocity
 The diver increases her
rotational velocity by pulling
into a tuck position, thus
reducing her rotational
inertia about her center of
gravity.
Kepler’s Second Law
 Kepler’s second law says
that the radius line from
the sun to the planet
sweeps out equal areas in
equal times.
 The planet moves faster in
its elliptical orbit when it is
nearer to the sun than
when it is farther from the
sun.
Kepler’s Second Law
 This is due to conservation
of angular momentum.
 The gravitational force
acting on the planet
produces no torque about
an axis through the sun
because the lever arm is
zero: the force’s line of
action passes through the
sun.
Kepler’s Second Law
 When the planet moves
nearer to the sun, its
rotational inertia about the
sun decreases.
 To conserve angular
momentum, the rotational
velocity of the planet about
the sun must increase.
Angular momentum is a vector
 The direction of the rotational-velocity vector is
given by the right-hand rule.
 The direction of the angular-momentum vector
is the same as the rotational velocity.
Inertia I, rotational velocity 
Angular momentum : L  I
A student holds a spinning bicycle wheel
while sitting on a stool that is free to
rotate. What happens if the wheel is
turned upside down?
To conserve
angular momentum,
the original direction
of the angularmomentum vector
must be maintained.
A student holds a spinning bicycle wheel
while sitting on a stool that is free to
rotate. What happens if the wheel is
turned upside down?
The angular
momentum of the
student and stool,
+Ls, adds to that of
the (flipped) wheel,
-Lw, to yield the
direction and
magnitude of the
original angular
momentum +Lw.
A student sits on a stool holding a bicycle wheel with a
rotational velocity of 5 rev/s about a vertical axis. The
rotational inertia of the wheel is 2 kg·m2 about its center and
the rotational inertia of the student and wheel and platform
about the rotational axis of the platform is 6 kg·m2. What is
the initial angular momentum of the system?
a)
b)
c)
d)
10 kg·m2/s upward
25 kg·m2/s downward
25 kg·m2/s upward
50 kg·m2/s downward
L = I = (2 kg·m2)(5 rev/s)
= 10 kg·m2/s
upward from plane of wheel