Transcript Chapter 10 - Dynamics of Rotational Motion

Chapter 10
Dynamics of
Rotational Motion
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
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Goals for Chapter 10
• To learn what is meant by torque
• To see how torque affects rotational motion
• To analyze the motion of a body that rotates as it
moves through space
• To use work and power to solve problems for
rotating bodies
• To study angular momentum and how it changes
with time
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Loosen a bolt
• Which of the three
equal-magnitude
forces in the figure
is most likely to
loosen the bolt?
• Fa?
• F b?
• Fc?
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Loosen a bolt
• Which of the three
equal-magnitude
forces in the figure
is most likely to
loosen the bolt?
• F b!
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Torque
• Let O be the point around
which a solid body will be
rotated.
O
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Torque
Line of
action
• Let O be the point around
which a solid body will be
rotated.
• Apply an external force F
on the body at some point.
O
• The line of action of a force
is the line along which the
force vector lies.
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F applied
Torque
Line of
action
• Let O be the point around
which a body will be
rotated.
• The lever arm (or moment
arm) for a force
L
O
is
The perpendicular distance
from O to the line of action
of the force
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F applied
Torque
Line of
action
• The torque of a force with
respect to O is the product of
force and its lever arm.
t=FL
L
O
• Greek Letter “Tau” = “torque”
• Larger torque if
• Larger Force applied
• Greater Distance from O
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F applied
Torque
t=FL
Larger torque if
• Larger Force applied
• Greater Distance from O
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Torque
Line of
action
• The torque of a force with
respect to O is the product of
force and its lever arm.
t=FL
L
O
• Units: Newton-Meters
• But wait, isn’t Nm = Joule??
F applied
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Torque
• Torque Units: Newton-Meters
• Use N-m or m-N
• OR...Ft-Lbs or lb - feet
• Not…
•
•
•
•
Lb/ft
Ft/lb
Newtons/meter
Meters/Newton
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Torque
• Torque is a
ROTATIONAL
VECTOR!
• Directions:
• “Counterclockwise” (+)
• “Clockwise” (-)
• “z-axis”
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Torque
• Torque is a
ROTATIONAL
VECTOR!
• Directions:
• “Counterclockwise”
• “Clockwise”
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Torque
• Torque is a
ROTATIONAL VECTOR!
• Directions:
• “Counterclockwise” (+)
• “Clockwise” (-)
• “z-axis”
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Torque
• Torque is a
ROTATIONAL
VECTOR!
• But if r = 0, no torque!
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Torque as a vector
• Torque can be expressed
as a vector using the
vector cross product:
t= r x F
• Right Hand Rule for
direction of torque.
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Torque as a vector
• Torque can be expressed
as a vector using the
vector cross product:
Line of
action
t= r x F
• Where r = vector from axis
of rotation to point of
application of Force
q = angle between r and F
• Magnitude: t = r F sinq
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Lever
arm
O
r
q
F applied
Torque as a vector
Line of
action
• Torque can be evaluated
two ways:
Lever
arm
t= r x F
t= r * (Fsin q)
O
r
F applied
remember sin q = sin (180-q)
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q
Fsinq
Torque as a vector
Line of
action
• Torque can be evaluated
two ways:
Lever
arm
r sin q
t= r x F
t= (rsin q) * F
The lever arm L = r sin q
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O
r
F applied
q
Torque as a vector
Line of
action
• Torque is zero three ways:
t= r x F
Lever
arm
r sin q
t= (rsin q) * F
t= r * (Fsin q)
If q is zero (F acts along r)
If r is zero (f acts at axis)
If net F is zero (other forces!)
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O
r
F applied
q
Applying a torque
• Find
t
if F = 900 N and q = 19 degrees
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Applying a torque
• Find t if F = 900 N
and q = 19 degrees
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Torque and angular acceleration for a rigid body
• The rotational analog of Newton’s second law for a
rigid body is tz =Iz.
• What is  for this example?
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Torque and angular acceleration for a rigid body
• The rotational analog of Newton’s second law for a
rigid body is tz =Iz.
t = I
and t = Fr
 = Fr/I
and a = r 
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so
I = Fr
Another unwinding cable – Example 10.3
• What is acceleration
of block and tension
in cable as it falls?
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Rolling as Translation and Rotation Combined
Identify that smooth rolling can be considered as a combination of
pure translation and pure rotation.
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Rolling as Translation and Rotation Combined


Assume objects roll smoothly (no slipping!)
The center of mass (com) of the object moves in a
straight line parallel to the surface

Object rotates around the com as it moves

Rotational motion is defined by:
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Rolling as Translation and Rotation Combined
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Rolling without slipping
• The condition for rolling without slipping is vcm = R.
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11-1 Rolling as Translation and Rotation Combined
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11-1 Rolling as Translation and Rotation Combined
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11-1 Rolling as Translation and Rotation Combined
Answer: (a) the same (b) less than
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Rigid body rotation about a moving axis
• Motion of rigid body is a
combination of translational
motion of center of mass and
rotation about center of mass
• Kinetic energy of a rotating &
translating rigid body is
KE = 1/2 Mvcm2 + 1/2 Icm2.
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Forces and Kinetic Energy of Rolling

Combine translational and rotational kinetic energy:
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The race of the rolling bodies
• Which one wins??
• WHY???
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A sphere smoothly rolls down a slope and accelerates…
• What are acceleration & magnitude of friction on ball?
• Use Newton’s second law for motion of center of mass
and rotation about center of mass.
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Acceleration of a rolling sphere

For smooth rolling down a ramp:
(If slip occurs, then the motion is not smooth rolling!)
1. Gravitational
force is vertically
down
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Acceleration of a rolling sphere

For smooth rolling down a ramp:
1. Gravitational
force is vertically
down
2. Normal force is
perpendicular to
ramp
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Acceleration of a rolling sphere

For smooth rolling down a ramp:
1. Gravitational
force is vertically
down
2. Normal force is
perpendicular to
ramp
3. Force of STATIC
friction points up
slope
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Acceleration of a rolling sphere
• Use Newton’s second law for the motion of the center
of mass and the rotation about the center of mass.
• What are acceleration & magnitude of friction on ball?
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Acceleration of a rolling sphere
• Use Newton’s second law for the motion of the center
of mass and the rotation about the center of mass.
• What are acceleration & magnitude of friction on ball?
Note that static frictional force produces the rotation
Without friction, the object will simply slide
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Answer: The maximum height reached by B is less than that
reached by A.
For A, all the kinetic energy becomes potential energy at h.
Since the ramp is frictionless for B,
all of the rotational K stays rotational,
and only the translational kinetic energy becomes
potential energy at its maximum height.
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A yo-yo
• What is the speed of the
center of the yo-yo after
falling a distance h?
• Is this less or greater than
an object of mass M that
doesn’t rotate as it falls?
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Acceleration of a yo-yo
• We have translation and rotation, so we use Newton’s
second law for the acceleration of the center of mass
and the rotational analog of Newton’s second law for
the angular acceleration about the center of mass.
•What is a and
T for the yo-yo?
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
As yo-yo moves down string, it loses
potential energy mgh but gains
rotational & translational kinetic energy
1. Rolls down a “ramp” of angle 90°
2. Rolls on an axle instead of its outer surface
3. Slowed by tension T rather than friction
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Example Calculate the acceleration of the yo-yo
o
o
M = 150 grams, R0 = 3 mm, Icom = Mr2/2 = 3E-5 kg m2
Therefore acom = -9.8 m/s2 / (1 + 3E-5 / (0.15 * 0.0032))
= - .4 m/s2
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Work and power in rotational motion
• Tangential Force over angle does work
• Total work done on a body by external torque is equal to the
change in rotational kinetic energy of the body
• Power due to a torque is P = tzz
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Work and power in rotational motion
• Electric Motor provide 10-Nm torque on
grindstone, with I = 2.0 kg-m2 about its shaft.
• Starting from rest, find work W down by
motor in 8 seconds and KE at that time.
• What is the average power?
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Work and power in rotational motion
• Analogy to 10.8: Calculating Power from Force
• Electric Motor provide 10-N force on block, with m =
2.0 kg.
• Starting from rest, find work W down by motor
in 8 seconds and KE at that time.
• What is the average power?
• W = F x d = 10N x distance travelled
• Distance = v0t + ½ at2 = ½ at2
• F = ma => a = F/m = 5 m/sec/sec => distance = 160 m
• Work = 1600 J; Avg. Power = W/t = 200 W
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Angular momentum
• The angular momentum of a rigid body rotating about a
symmetryaxis is parallel to the angular velocity and is
given by L = I.
• Units = kg m2/sec (“radians” are implied!)
• Direction = along RHR vector of angular velocity.
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Angular momentum
• For any system of particles t = dL/dt.
• For a rigid body rotating about the z-axis tz = Iz.
Ex 10.9: Turbine with I = 2.5 kgm2;  = (40 rad/s3)t2
What is L(t) & L @ t = 3.0 seconds; what is t(t)?
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Angular Momentum


Angular momentum has meaning only with respect to a
specified origin and axis.
L is always perpendicular to plane formed by position &
linear momentum vectors
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Angular Momentum
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11-5 Angular Momentum
a) Rank particles according to magnitudes of Angular Momentum about O
1 & 3, 2 & 4, 5
b) Which have negative ang. Momentum about O?
2 and 3 (assuming counterclockwise is positive)
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Conservation of angular momentum
• When the net external torque acting on a system is zero, the total
angular momentum of the system is constant (conserved).
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A rotational “collision”
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Angular momentum in a crime bust
• A bullet (mass = 10g, @ 400 m/s) hits a door (1.00 m wide
mass = 15 kg) causing it to swing. What is  of the door?
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Angular momentum in a crime bust
• A bullet (mass = 10g, @ 400 m/s) hits a door (1.00 m wide
mass = 15 kg) causing it to swing. What is  of the door?
• Lintial = mbulletvbulletlbullet
• Lfinal = I = (I door + I bullet) 
• Idoor = Md2/3
• I bullet = ml2
• Linitial = Lfinal so mvl = I
• Solve for  and KE final
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Gyroscopes and precession
• For a gyroscope, the axis of rotation changes direction.
The motion of this axis is called precession.
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Gyroscopes and precession
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Gyroscopes and precession
• For a gyroscope, the axis of
rotation changes direction.
The motion of this axis is
called precession.
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A rotating flywheel
• Magnitude of angular momentum constant, but its direction
changes continuously.
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A precessing gyroscopic
• Example 10.13
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