Download Rotational Dynamics

Rotational Dynamics
Chapter 9
Expectations
After Chapter 9, students will:
 calculate torques produced by forces
 recognize the condition of complete equilibrium
 calculate the location of the center of gravity of a
collection of objects
 use the rotational form of Newton’s second law of
motion to analyze physical situations
 calculate moments of inertia
Expectations
After Chapter 9, students will:
 calculate the rotational work done by a torque
 calculate rotational kinetic energy
 calculate angular momentum
 apply the principle of the conservation of angular
momentum in an isolated system
Preliminary Definitions

Torque

Complete Equilibrium

Center of Gravity
Torque
Torque: the rotational analog to force
Force produces changes in linear motion (linear
acceleration). A force is a push or a pull.
Torque produces changes in angular motion
(angular acceleration). A torque is a twist.
Torque
length of lever arm
Mathematical definition:
  Fl
SI units: N·m
torque
force
The lever arm is the line
through the axis of
rotation, perpendicular to
the line of action of the
force.
Torque
Torque is a vector quantity. It magnitude is given
by
  Fl
and its direction by the right-hand rule:
F
l
F
torque vector
points out of page
rotate l into F
l
Torque
  Fl
For a given force, the torque depends on the
location of the force’s application to a rigid
object, relative to the location of the axis of
rotation.
more torque
less torque
Torque
  Fl
For a given force, the torque depends on the force’s
direction.
Complete Equilibrium
A rigid object is in complete equilibrium if the sum
of the forces exerted on it is zero, and the sum of
the torques exerted on it is zero.
F
x
0
F
y
0
  0
An object in complete equilibrium has zero
translational (linear) acceleration, and zero
angular acceleration.
Center of Gravity
In analyzing the equilibrium of an object, we see
that where a force is applied to an object
influences the torque produced by the force.
In particular, we sometimes need to know the
location at which an object’s weight force acts on
it.
Think of the object as a collection of smaller pieces.
Center of Gravity
In Chapter 7, we calculated the location of the
center of mass of this system of pieces:
m1 x1  m2 x2  ...  mi xi
xC 
m1  m2  ...  mi
Multiply numerator and denominator by g:
m1 gx1  m2 gx2  ...  mi gxi
xC 
m1 g  m2 g  ...  mi g
Center of Gravity
m1 gx1  m2 gx2  ...  mi gxi
xC 
m1 g  m2 g  ...  mi g
But: W  mg
Substituting:
W1 x1  W2 x2  ...  Wi xi
xC 
W1  W2  ...  Wi
It is intuitive that the weight force acts at the
effective location of the mass of an object.
Newton’s Second Law: Rotational
Consider an object, mass m, in circular motion with
a radius r. We apply a tangential force F:
F
The result is a
tangential acceleration
F  maT
according to Newton’s second law.
r
Newton’s Second Law: Rotational
The torque produced by the force is   Fr  maT r
But the tangential acceleration
is related to the angular
acceleration: aT  r
Substituting:
F
  mr r  mr 
2
r
Newton’s Second Law: Rotational
  mr 
2
This is an interesting result.
If we define the quantity
I  mr
2
as the moment of inertia,
r
we have   I
the rotational form of Newton’s second law.
F
Moment of Inertia
The equation I  mr
gives the moment of inertia of a “particle” (meaning
an object whose dimensions are negligible
compared with the distance r from the axis of
rotation).
2
Scalar quantity; SI units of kg·m2
Moment of Inertia
Not many real objects can reasonably be
approximated as “particles.” But they can be
treated as systems of particles …
I  m1r1  m2 r2  ...  mi ri
2
I   mi ri
i
2
2
2
Moment of Inertia
The moment of inertia of an object depends on:
 the object’s total mass
 the object’s shape
 the location of the axis of rotation
Rotational Work and Energy
By analogy with the corresponding translational
quantities:
Translational
Rotational
W  Fs cos 
WR  
SI units: N·m = J
1 2
KE  mv
2
1 2
KER  I
2
SI units: (kg·m2) / s2 = N·m = J
Total Mechanical Energy
We now add a term to our idea of the total
mechanical energy of an object:
1 2 1 2
E  mv  I  mgh
2
2
total energy
gravitational
potential energy
translational
rotational
kinetic energy
kinetic energy
Angular Momentum
By analogy with linear momentum:
p  mv
L  I
Angular momentum is a vector quantity. Its
magnitude is given by
L  I
SI units: kg·m2 / s
and its direction is the same as the direction of .
 must be expressed in rad/s.
Angular Momentum: Conservation
If a system is isolated (no external torque acts on it),
its angular momentum remains constant.
[If a system is isolated (no external force acts on it),
its linear momentum remains constant.]