rotational dynamics chapter 9. expectations after chapter 9, students will: calculate torques...
Post on 18-Dec-2015
221 Views
Preview:
TRANSCRIPT
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
Mathematical definition:
The lever arm is the line
through the axis of
rotation, perpendicular to
the line of action of the
force.
Fltorque
force
length of lever arm
SI units: N·m
Torque
Torque is a vector quantity. It magnitude is given by
and its direction by the right-hand rule:
Fl
F
l l
Frotate l into F
torque vectorpoints out of page
Torque
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.
Fl
more torque less torque
Torque
For a given force, the torque depends on the force’s direction.
Fl
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.
An object in complete equilibrium has zero translational (linear) acceleration, and zero angular acceleration.
0 0 0 yx FF
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:
Multiply numerator and denominator by g:
i
iiC mmm
xmxmxmx
...
...
21
2211
gmgmgm
gxmgxmgxmx
i
iiC
...
...
21
2211
Center of Gravity
But: Substituting:
It is intuitive that the weight force acts at the effective location of the mass of an object.
gmgmgm
gxmgxmgxmx
i
iiC
...
...
21
2211
mgW
i
iiC WWW
xWxWxWx
...
...
21
2211
Newton’s Second Law: Rotational
Consider an object, mass m, in circular motion with a radius r. We apply a tangential force F:
The result is a
tangential acceleration
according to Newton’s second law.
r
F
TmaF
Newton’s Second Law: Rotational
The torque produced by the force is
But the tangential acceleration
is related to the angular
acceleration:
Substituting:r
F
2mrrrm
rmaFr T
raT
Newton’s Second Law: Rotational
This is an interesting result.
If we define the quantity
as the moment of inertia,
we have
the rotational form of Newton’s second law.
r
F
2mr
2mrI
I
Moment of Inertia
The equation
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).
Scalar quantity; SI units of kg·m2
2mrI
Moment of Inertia
Not many real objects can reasonably be approximated as “particles.” But they can be treated as systems of particles …
iii
ii
rmI
rmrmrmI
2
2222
211 ...
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
2
2
1
cos
mvKE
FsW
2
2
1
IKE
W
R
R
SI units: N·m = J
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:
mghImvE 22
2
1
2
1
total energy
translational
kinetic energy
rotational
kinetic energy
gravitational
potential energy
Angular Momentum
By analogy with linear momentum:
Angular momentum is a vector quantity. Its magnitude is given by
and its direction is the same as the direction of .
must be expressed in rad/s.
ILmvp
IL SI units: kg·m2 / 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.]
top related