Download Angular velocity

Angular velocity
Angular velocity
measures
how quickly the object is rotating.
Average angular velocity
Instantaneous angular velocity
Two coins rotate on a turntable. Coin B is twice as
far from the axis as coin A.
A.
The angular velocity of A is twice that of B.
B.
The angular velocity of A equals that of B.
C. The angular velocity of A is half that of B.
Two coins rotate on a turntable. Coin B is twice as far from
the axis as coin A.
A.
The speed of A is twice that of B.
B.
The speed of A equals that of B.
C. The speed of A is half that of B.
Angular Acceleration
Angular acceleration
measures how rapidly the
angular velocity is changing:
Average angular acceleration
Instantaneous angular acceleration
Linear motion definitions versus
rotational motion definitions
(average values)
Linear kinematics vs. rotational kinematics
A bicycle travels 150 m along a circular track of radius
15 m. What is the angular displacement in radians of
the bicycle from its starting position?
A wheel with a 0.10-m radius is rotating at 30 rev/s.
It then slows uniformly to 15 rev/s over a 3.0 second
interval. What is the angular acceleration of a point
on the wheel?
q
A graph of angle versus time appears like so:
Which of the following angular velocity graphs matches the
above angle graph?
w
w
A.
w
B.
w
C.
D.
w
A graph of angular velocity versus time for a
rolling wheel appears like so:
Which of the following angle graphs matches the above angular
velocity graph?
A.
B.
C.
D.
Sign of the Angular Acceleration
A ladybug sits at the outer edge of a merry-go- round, and a
gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution once
each second. The gentleman bug’s angular speed is
1. half the ladybug’s.
2. the same as the ladybug’s.
3. twice the ladybug’s.
4. impossible to
determine
Example
A high-speed drill rotating CCW takes 2.5 s to speed up to 2400 rpm
from rest.
A. What is the drill’s angular acceleration?
B. How many revolutions does it make as it reaches top speed?
Interpretation of Torque
Torque is due to the component of the force perpendicular to the
radial line.
  rF  rF sin 
A Second Interpretation of Torque
  r F  rF sin 
Which factor does the torque on an object not depend on?
A.
B.
C.
D.
The magnitude of the applied force.
The object’s angular velocity.
The angle at which the force is applied.
The distance from the axis to the point at which the
force is applied.
A net torque applied to an object causes
A.
a linear acceleration of the object.
B.
the object to rotate at a constant rate.
C. the angular velocity of the object to change.
D. the moment of inertia of the object to change.
The four forces shown have the same strength. Which force
would be most effective in opening the door?
A.
B.
C.
D.
E.
Force F1
Force F2
Force F3
Force F4
Either F1 or F3
A string is tied to a doorknob 0.80 m from the hinge as shown in
the figure. At the instant shown, the force applied to the string is
5.0 N. What is the torque on the door?
hinge
Newton’s Second Law for Rotation
I = moment of inertia. Objects with larger moments of inertia are
harder to get rotating.
Which has a greater rotational
inertia about its center of mass?
1.
2.
3. They are equal
You have two uniform solid cylinders of the same
material. Cylinder 2 has twice the radius of
cylinder 1 but the same length. By what factor
does the CM moment of inertia (about symmetry
axis) I2 of cylinder 2 exceed the moment of inertia
I1 of cylinder 1.
(1) 4
(2) 8
(3) 16
(4) their moments of inertia are the same.
Moments of Inertia of Common Shapes
Rolling Is a Combination of Translation and Rotation
Rotational Kinetic Energy
The kinetic energy of rotating point mass, m1, is given by
For several points rotating, the total kinetic energy is the sum of the kinetic energy
of all point masses
We know that all points in a rigid body rotate at the same angular velocity,
w i = wj = w, where i = 1, 2, . . . ,N and j = 1, 2, . . . ,N. Therefore,
Rotational Kinetic Energy (continued)
Recall that the moment of inertia was given by the expression,
We can now substitute in the expression for the moment of inertia of the object into
our rotational kinetic energy equation to get,
The total kinetic energy of an object is the sum of the translational kinetic energy of
the object’s center-of-mass and the rotational kinetic energy about the object’s
center-of-mass.
Angular Momentum
Similar to the linear momentum of a moving object, a rotating object has angular
momentum,
For a closed system, angular momentum is conserved,
Using the above expressions, we may rewrite the conservation of momentum
equation as
A student sits on a rotating stool holding two weights, each of mass
5.00 kg and approximated as point masses. When his arms are extended
horizontally, the weights are 0.850 m from the axis of rotation and the
student rotates with an angular speed of 2.00 rad/s. The moment of
inertia of the student plus stool is a constant, 3.00 kg·m2. While
spinning, the student pulls the weights inward horizontally to a position
0.250 m from the axis of rotation.
Calculate the angular speed of the
student after the weights have been
pulled inward.