ROTATIONAL MOTION

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ROTATIONAL MOTION
AND EQUILIBRIUM
Angular Quantities of Rotational Motion
Rotational Kinematics Torque Center of Gravity
Moment of Inertia Rotational Kinetic Energy
Angular Momentum Rotational Equilibrium
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ANGULAR CONVERSIONS
p
1 rev 2
rad 360 deg
p
rad 180 deg
Convert 246
o
to radians.
Convert 16.4 rev to degrees.
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Convert 246
o
to radians.
Convert 16.4 rev to degrees.
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ROTATIONAL KINEMATICS
Uniform angular acceleration
q
instantaneous angular position
w
instantaneous angular speed
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EXAMPLE
Through what angle does a wheel rotate if it
begins from rest
and accelerates at 3
o
/s
2
for 15 seconds? Also calculate its final
angular speed.
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ANSWER
Through what angle does a wheel rotate if it
begins from rest
and accelerates at 3
o
/s
2
for 15 seconds? Also calculate its final
angular speed.
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TORQUE Torque, t, is produced when a force acts
on a body along a line that is displaced from a
pivot point or axis of rotation. LINE OF ACTION
The line of action of a force is the line along
which the force acts. LEVER ARM The lever arm
of a force is the distance from the pivot or axis
of rotation to the point where the force is
applied.
The torque produced by this force is trying to
rotate the bar in a counter-clockwise direction.
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HOW TO CALCULATE A TORQUE

• 1. In a sketch, locate the pivot point or axis of
  rotation.
• Locate the force producing the torque and where
  it is applied.
• 3. Determine the length of the line segment from
  the pivot to the point where the force is
  applied. This is the lever arm.
• 4. Determine the angle between the force and the
  lever arm. The force multiplied by the sine of
  this angle is the force component perpendicular
  to the lever arm.
• The magnitude of the torque is the product of the
  force perpendicular component and the lever arm.
• Determine the direction of rotation the torque is
  trying to induce as being either clockwise or
  counter-clockwise.

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.
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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.
1. Identify the pivot (left end of rod)

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.

1. Identify the pivot (left end of rod)
2. Identify the force ( F 16 N)

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.

1. Identify the pivot (left end of rod)
2. Identify the force ( F 16 N)
3. Determine the lever arm ( r 1.25 m)

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.

• Identify the pivot (left end of rod)
• Identify the force ( F 16 N)
• Determine the lever arm ( r 1.25 m)
• Determine the angle between the force and the
  lever arm (q 57o)

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.

• Identify the pivot (left end of rod)
• Identify the force ( F 16 N)
• Determine the lever arm ( r 1.25 m)
• Determine the angle between the force and the
  lever arm (q 57o)
• Calculate the torque

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EXAMPLE
Calculate the torque produced by
the force acting on the rod as shown.

• Identify the pivot (left end of rod)
• Identify the force ( F 16 N)
• Determine the lever arm ( r 1.25 m)
• Determine the angle between the force and the
  lever arm (q 57o)
• Calculate the torque
• Determine the orientation of the torque. (see
  diagram)

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When g is considered constant cg is usually
referred to as center of mass.
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EXAMPLE
y
Suppose the three masses in the diagram


m
4 kg
at (0, 3m)
are fixed together by massless rods and are
2
free to rotate in the xy plane about their
r
2
center of gravity. The coordinates of their
center of gravity are (1m, 1m) as calculated
r
r
m
3 kg at (4m, 0)
3
1
3
in the previous example.
x
Calculate the moment of inertia of these
m
5 kg at (0, 0)
masses about this axis.
1
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ANSWER
y
Suppose the three masses in the diagram


m
4 kg
at (0, 3m)
are fixed together by massless rods and are
2
free to rotate in the xy plane about their
r
2
center of gravity. The coordinates of their
center of gravity are (1m, 1m) as calculated
r
r
m
3 kg at (4m, 0)
3
1
3
in the previous example.
x
Calculate the moment of inertia of these
m
5 kg at (0, 0)
masses about this axis.
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EXAMPLE A solid ball rolls down a 40o incline
from a height of 4m without slipping. The radius
of the ball is 20cm and it begins from rest. What
is the linear speed of the ball at the bottom of
the incline?
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ANSWER A solid ball rolls down a 40o incline
from a height of 4m without slipping. The radius
of the ball is 20cm and it begins from rest. What
is the linear speed of the ball at the bottom of
the incline?
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Continues from previous slide
Substitutions made to produce working equation
Simplify and solve for v
The speed of the ball does not depend on its mass
nor on its radius. It even does not depend on the
angle of the incline, just the height from which
it starts. Do the same calculation with a
cylinder and determine which will reach the
bottom of the incline first, the sphere or the
cylinder if released together.
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EXAMPLE A merry-go-round on a playground has a
radius or 1.5 m and a mass of 225 kg. One child
is sitting on its outer edge as it rotates 1
rev/s. If the mass of the child is 50 kg, what
will the new rotation rate be when the child
crawls half way to the center of the
merry-go-round?
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ANSWER A merry-go-round on a playground has a
radius or 1.5 m and a mass of 225 kg. One child
is sitting on its outer edge as it rotates 1
rev/s. If the mass of the child is 50 kg, what
will the new rotation rate be when the child
crawls half way to the center of the
merry-go-round?
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An object in equilibrium has no linear and no
angular accelerations. That does not mean it
isnt moving, just not accelerating. When an
object is stationary it is said to be in static
equilibrium. Many of the equilibrium problems
here will be static equilibrium problems.
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Solution on next several slides
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STEP 1
cable
wall
T
rod
o
45
o
30
hinge
R
W
Add Force Vectors T Tension, W Weight, R
Reaction
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STEP 1
cable
wall
rod
45
o
pivot
o
30
hinge
Add Force Vectors T Tension, W Weight, R
Reaction Add needed angles.
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STEP 1
Add Force Vectors T Tension, W Weight, R
Reaction Add needed angles. Add Lever Arms.
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On the next slide these force equations will be
used to solve for Rx and Ry.
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STEP 3 Torque Analysis
To analyze the torques acting on the rod,
construct to following table or small
spreadsheet. The column headings are
The entries in each row correspond to values
associated with one force on the rod. One row
for each force. The next slide continues the
example solution.
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g 9.8 m/s2 was used