Dynamics of Rotational Motion

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Dynamics of Rotational Motion

• Torque the rotational analogue of force
• Torque force x moment arm
• t Fl moment arm perpendicular distance
  through which the force acts
• a.k.a. lever arm

l
l
F
F
l
l
F
F
t Fl F r sin f Ftan r
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Example To loosen a difficult nut, a wrench with
a piece of pipe for additional leverage is used.
If the distance from the center of the nut to the
end of the pipe is 0.80 m, and the wrench and
pipe make an angle of 20º with respect to the
horizontal, determine the magnitude and direction
of the torque applied by a 900 N man standing on
the edge of the pipe.
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Torque and Angular Acceleration single particle
contribution
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• Problem solving tactics
• Draw a diagram
• For each body, draw a free body diagram
• Choose coordinates and directions (including for
  rotations)
• Relate torques to angular accelerations and
  forces to linear accelerations
• for more than one body, repeat and note any
  geometrical relation between motions of bodies

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A cord is wrapped around a solid 50 kg cylinder
which has a diameter of 0.120 m, and which
rotates (frictionlessly) about an axis through
its center. A 9.00 N force is applied to the end
of the cable, causing the cable to unwind and the
drum (initially at rest) to rotate. What is the
angular acceleration of the drum?
9.00 N
A mass m is suspended by a string wrapped around
a pulley of radius R and moment of inertia I.
The mass and pulley are initially at rest.
Determine the acceleration of the disk and of the
suspended mass in terms of the given parameters
(m, I, R). Examine the special case where the
pulley is a uniform disk of mass M.
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A mass m2 is suspended by a string wrapped around
a pulley of radius R and moment of inertia I.
The string is then connected to a mass m1 which
slides on a horizontal frictionless surface. The
masses and pulley are initially at rest.
Determine the accelerations of the masses and the
pulley in terms of the the given parameters (m1,
m2, I, R). Examine the special case where the
pulley can be treated as a thin cylindrical shell
of mass M.
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y-axis cm axis axis of rotation
More on Combining Translation and Rotation
F1,y
vi
vcm
ri
vi
vcm
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example A primitive yo-yo is made of a solid
cylinder of mass M with string wrapped around its
radius R. What is the speed of the cylinder after
it has dropped a distance h? What is acceleration
of the cylinder and the tension in the string?
example A solid bowling bowl rolls without
slipping down a ramp inclined at an angle b with
respect to the horizontal. What is the balls
acceleration?
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Rolling Friction deforming surfaces
Distorted surface normal force offset from cm,
net torque slows rotation
Stiff surface normal force through cm, no torque
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Work and Power in Rotational Motion Force applied
to a point on a rotating object
Ftan
s
R
q
Ftan
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If the power output of an engine is 200 hp at
6000rpm, what is the corresponding torque?
A motor produces a torque of 10 N-m with disk
mounted to its shaft. The disk has a moment of
inertia of 2.0 kg m2. If the system starts from
rest, determine the work done by the motor in 8.0
s, and the kinetic energy of the disk at the end
of this time. What is the average power
delivered during this 8 seconds?
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Angular momentum for a single particle, defined
as L ? r p r mv depends upon origin! L
mvr sinf mvr? mv?r
mv
f
r?
w
Right hand rule for direction
Generalize to composite object In the absence of
external torques, angular momentum is conserved
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Two disks with moments of inertia Ia and Ib share
a common axis of rotation, but have different
angular velocities (wa and wb ). The disks are
brought together so that they eventually reach a
common final angular velocity. Derive an
expression for that final angular velocity w.
Suppose the two disks are uniform with masses of
2.0 kg and 4.0 kg, radii of .20 m and .10 m and
initial angular speeds of 50 rad/s and 200 rad/s.
Determine the final angular speed and the change
in the kinetic energy of the system.
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A bullet (10 g and initial speed 400 m/s) is
fired into a door from a direction perpendicular
to the door. The door has a mass of 15 kg and is
1.0 m wide. Determine the angular velocity of
the door after the bullet imbeds itself in the
center of the door.
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Gyroscopes and Precession
t
L
w
w