Chapter 11
Rotational Dynamics and
Static Equilibrium
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Units of Chapter 11
• Torque
• Torque and Angular Acceleration
• Zero Torque and Static Equilibrium
• Center of Mass and Balance
• Dynamic Applications of Torque
• Angular Momentum
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Units of Chapter 11
Optional
Not required
• Conservation of Angular Momentum
• Rotational Work and Power
• The Vector Nature of Rotational Motion
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11-1 Torque
From experience, we know that the same force
will be much more effective at rotating an
object such as a nut or a door if our hand is not
too close to the axis.
This is why we have
long-handled
wrenches, and why
doorknobs are not
next to hinges.
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11-1 Torque
We define a quantity called torque:
The torque increases as the force increases,
and also as the distance increases.
r is moment arm length
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11-1 Torque
Only the tangential component of force causes
a torque:
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11-1 Torque
This leads to a more general definition of torque:
is called moment arm length, it’s the perpendicular
distance between the rotation center to the force line.
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r is the moment arm length.
To use longer r, can saves force.
Examples: Level, bottle opener, dolly, wrench, screw driver….
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11-1 Torque
If the torque causes a counterclockwise angular
acceleration, it is positive; if it causes a
clockwise angular acceleration, it is negative.
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11-3 Zero Torque and Static Equilibrium
Static equilibrium occurs when an object is at
rest – neither rotating nor translating.
Total torque is zero respect to any rotational axis.
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11-3 Zero Torque and Static Equilibrium
If the net torque is zero, it doesn’t matter which
axis we consider rotation to be around; we are
free to choose the one that makes our
calculations easiest.
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|τccw| = |τcw|
When you consider
SIZE only, use the
absolute value.
CCW torque=CW torque
When you consider
total torque = 0,
Ccw torque are positive
Cw torque are negative.
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1, label forces,
2, choose rotational axis
3, write out torque created by each forces.
Use the correct moment arm length for each force.
If a force is going through the rotational axis, its torque=0
4, use equation total torque = 0, To solve bicep force,
5, Finally use equation total forces =0, to solve the
unknown bone force. Will the bicep force be more or less
than Mg+mg?
Will the bone force point upward or
downward?
What if the bone becomes a soft rope?
Equations:
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11-4 Center of Mass and Balance
If an extended object is to be balanced, it must
be supported through its center of mass.
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11-4 Center of Mass and Balance
This fact can be used to find the center of mass
of an object – suspend it from different axes and
trace a vertical line. The center of mass is where
the lines meet.
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11-2 Torque and Angular Acceleration
Newton’s second law:
If we consider a mass m rotating around an
axis a distance r away, we can reformat
Newton’s second law to read:
Or equivalently,
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11-2 Torque and Angular Acceleration
Once again, we have analogies between linear
and angular motion:
=∑mr2
= at/r
= Fr
s
v
p=mv
KE=1/2 mv2
θ =s /r
ω =vt /r
L =I ω
KE =1/2 I ω2
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Example: open your arm and let it fall without applying any muscle force.
What will be the angular acceleration for your arm due to its
gravity at that moment? What will be the linear acceleration of
your finger tip at that moment? Will your finger tip fall faster,
or slower than free fall? put a small object at your finger tip to
compare. Will your upper arm fall faster or slower than free
fall? The length of the whole arm = L; Total arm mass = m
Choose the shoulder to be the rotation axis,
Total torque exerted on the arm:
L
Assume the whole arm as a long rod of block,
When rotating around its end(the shoulder)
mg
Arm’s rotational inertia:
At that moment,the angular acceleration :
Fingertip linear acceleration:
Arm center linear acceleration:
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Fingertip fall (rotates) faster than Copyright
free fall.
Upper arm falls slower than free fall.
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11-5 Dynamic Applications of Torque
When dealing with systems that have both
rotating parts and translating parts, we must be
careful to account for all forces and torques
correctly.
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11-6 Angular Momentum
Using a bit of algebra, we find for a
particle moving in a circle of radius r,
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11-7 Conservation of Angular Momentum
If the net external torque on a system is zero,
the angular momentum is conserved.
The most interesting consequences occur in
systems that are able to change shape:
If < Ii
So that, ωf > ωi
As the moment of
inertia decreases, the
angular speed
increases, so the
angular momentum
does not change. 26
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Summary of Chapter 11
• A force applied so as to cause an angular
acceleration is said to exert a torque.
• Torque due to a tangential force:
• Torque in general:
r is the moment arm length. Longer r, saves force.
(but longer r means to apply force across longer
distance. It saves forces, but nothing can save work.
• Newton’s second law for rotation:
Angular acceleration = total torque / rotational inertia
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Once again, we have analogies between linear
and angular motion:
=∑mr2
= at/r
= Fr
s
v
p=mv
KE=1/2 mv2
θ =s /r
ω =vt /r
L =I ω
KE =1/2 I ω2
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Summary of Chapter 11
• In order for an object to be in static equilibrium, the
total force and the total torque acting on the object
must be zero. (most important)
• An object balances when it is supported at its center
of mass.
• In systems with both rotational and linear
motion, Newton’s second law must be applied
separately to each.
• Angular momentum:
•In systems with no external torque, total
angular momentum is conserved.
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11-7 Conservation of Angular Momentum
Optional
Planet’s elliptical orbit:
Rotating around the sun
Gravity of the force going through the rotational
center (the sun) hence doesn’t create torque on the
planet to change its angular momentum.
Iω will be constant, hence mr2ω will be constant.
After the same amount of time t, the area swept by
the line between a planet and the sun
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Area=r2∆θ=r2ωt, will be the same.
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11-7 Conservation of Angular Momentum
Optional
Angular momentum is also conserved in
rotational collisions:
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11-6 Angular Momentum
Optional
Looking at the rate at which angular momentum
changes,
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11-8 Rotational Work and Power
Optional Not Require
A torque acting through an angular
displacement does work, just as a force
acting through a distance does.
The work-energy theorem applies as
usual.
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11-8 Rotational Work and Power
Optional Not Require
Power is the rate at which work is done, for
rotational motion as well as for translational
motion.
Again, note the analogy to the linear form:
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11-9 The Vector Nature of Rotational
Motion, Optional Not Require
The direction of the angular velocity vector is
along the axis of rotation. A right-hand rule
gives the sign.
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11-9 The Vector Nature of Rotational
Motion, Optional Not Require
Conservation of angular momentum means that
the total angular momentum around any axis
must be constant. This is why gyroscopes are
so stable.
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