Physics 2211 Mechanics Lecture 20 OneDimensional Rotation

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Physics 2211 - MechanicsLecture
20One-Dimensional Rotation

• Dr. John Evans

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Plotting a and w
The angular acceleration a is the rate of
change of the angular velocity w. If the
angular acceleration a and the angular velocity w
are plotted vs. time t, then a is the slope of
the w vs. t curve, and Dw w(tf)-w(ti) is the
area under the a vs. t curve in the time interval
between ti and tf.
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Example A Rotating Wheel
Figure (a) shows the angular velocity w vs.
time t. What rotational behavior does it
describe, and what is the angular acceleration a?
The angular velocity w is positive, so the
plot describes a counterclockwise rotation that
increases from t0 to t1, remains constant from
tt1 to t2, and decreases to a halt from tt2 to
t3. Figure (b) shows the corresponding
angular acceleration a.
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Rotational vs. Linear Kinematics
Angular acceleration a is constant
Acceleration a is constant
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Example A Rotating Crankshaft
A cars tachometer indicates the angular
velocity w of the crank shaft in rpm. A car
stopped at a traffic light has its engine idling
at 500 rpm. When the light turns green, the
crankshafts angular velocity speeds up at a
constant rate to 2500 rpm in a time interval of
3.0 s. How many revolutions does the
crankshaft make in this time interval?
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Clicker Question 1
The fan blade shown is slowing down. Which
option describes a and w?
(c) wlt0 and agt0
(d) wlt0 and alt0.
(a) wgt0 and agt0
(b) wgt0 and alt0
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Rotation About a Center of Mass
If an object consists of N particles, all of the
same mass m, then the x equation becomes
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Example The Center of Mass

• A 500 g ball and a2 kg ball are
  connectedby a massless 50 cm rod.
• Find the center ofmass of the system
• What is the speed of each ball if they rotate
  around the center of mass at 40 rpm?

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Calculating the Center of Mass
Let the cell segmentation become very small,
so that Dm dm 0.
For any symmetrical object of uniform
density, the center of mass is located at the
geometrical center of the object.
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ExampleThe Center of Mass of a Rod
Find the center of mass of a thin uniform
rod of length L and mass M. Find the
tangential acceleration of one tip of a 1.6 m rod
that rotates about its center of mass with an
angular speed of 6.0 rad/s2.
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The Center of Rotation
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Torque

• The ability of a force to cause a
  rotation or twisting motion depends on three
  factors
• The magnitude F of the force
• The distance r from the point of application to
  the pivot
• The angle f at which the force is applied.

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Sign of Torque
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Two Interpretations of Torque
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Clicker Question 2
Five different forces are applied to the
same rod, which pivots around the black dot.
Which force produces the smallest torque about
the pivot?
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Net Torque
The object is free to rotate, but the axle
is fixed and prevents translation. The axle will
produce a force Faxle that is equal and opposite
to the sum of the other forces, so that Fnet0.
Each of the forces will produce a torque, and
the net torque about the axel is the sum of the
torques due to the applied forces
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Gravitational Torque
An object fixed on a pivot (taken as the
origin) will experience gravitational forces that
will produce torques. The torque about the pivot
from the ith particle will be ti-ximig. The
minus sign is because particles to the right of
the origin (x positive) will produce clockwise
(negative) torques.
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Balance
An object balances when the pivot point is
located directly under (or over) the center of
mass. For that situation, tgrav0.
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Example The Gravitational Torque on a Beam
The 4.0 m long 500 kg steel beam is supported
1.20 m from the right end. What is the
gravitational torque about the support?
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Couples
One way of producing rotation without
translation of an object is to apply a pair of
equal and opposite forces with offset lines of
action. Consider the situation shown.
Forces F1 and F2 are equal in magnitude, opposite
in direction, and their lines of action are
offset by distance l. The moment arm of F1,2 is
d1,2, and d1d2l. Then the net torque about
the pivot point is
The net torque does not depend on the
location of the pivot point. Therefore, the
couple exerts the same torque tlF about any
point on the object.
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Rotational Dynamics
Therefore, torque causes angular
acceleration. The relation is analogous to Fma.
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Newtons 2nd Law for Rotation
A rigid body undergoes pure rotational
motion about a fixed, frictionless, and unmoving
axis.
Definition Moment of Inertia
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Rotational and Linear Dynamics
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Example Rotating Rockets
Far out in space, a 100,000 kg rocket and a
200,000 kg rocket are docked at opposite ends of
a motionless 90 m long connecting tunnel. The
tunnel is rigid and its mass is much less than
that of either rocket. The rockets start their
engines simultaneously and fire in opposite
directions, each generating 50,000 N of thrust.
What is the angular velocity w of the structure
after 30 s?
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Clicker Question 3
Which system has the largest angular acceleration?
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Moment of Inertia
For discrete masses, the moment of inertia is
What about I for solid objects? Divide up
the object into cells of mass Dm, each cell with
coordinates (xi,yi), with ri2xi2yi2 and the
origin (0,0) taken as the rotation axis.
Now let the cells become infinitesimally
small. Then the sum becomes an integral over the
volume of the object.
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ExampleI of a Rod about a Pivot
Find the moment of inertia for a thin,
uniform rod of length L and mass M that rotates
about a pivot point at one end.
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Example I of a Circular Disc
Find the moment of inertia for a circular disc of
radius R and mass M that rotates on an axis
through its center.
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Moments of Inertia of Objects
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The Parallel-Axis Theorem
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Clicker Question 4
The T-shaped object is rotated about each axis
shown. For which axis does it have the largest
moment of inertia?
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End of Lecture 20

• Before the next lecture (11/28), read Knight,
  Sections 13.7 through 13.10.