Transcript Rotational or Angular Motion

Rotational or Angular Motion
When an object’s center of mass
moves from one place to another,
we call it linear motion or
translational motion.
Rotational motion (also called
angular motion) describes the
motion of an object around a
fixed line called the “axis of
rotation” or around a fixed point
called the “fulcrum” or “pivot”.
Suppose we have a wheel that is
attached to wall at its center so
that it can spin (sort of like the
“Wheel of Fortune”) around an
axle.
Will a force cause it to spin?
This force will NOT cause it to
spin…because it is directed through
the pivot or axle.
Will a force directed inward on
the wheel cause it to spin?
Push straight
into the page on
the center.
No, a force straight inward on the pivot (green
dot) will NOT cause it to spin, because the
force is directed through the pivot.
So…what will cause the wheel to
spin?
R
F
If we direct the force so that it does NOT go through the pivot,
then it WILL cause the wheel to spin.
This force is a distance, R, from the center. We call this
distance the “lever arm”…..and we call this type of force a
torque.
An object will only rotate (or spin) if a torque
is applied to it.
A torque is a force that does NOT go through
the axle or pivot.
Therefore, a torque has a lever arm:
 = FR
To calculate the torque, multiply
the force applied times the
perpendicular distance from the
line of force to the pivot.
Suppose we apply a force of 10 newtons
to the edge of the wheel, which has a
radius of 0.5 meters. What is the size of
the torque?
R
F
Answer: Since the force and radius are already
perpendicular, the torque is just the product of the
two.  = (10 N)(0.5 m) = 5 N-m
Now, calculate the torque when the force
is applied at a 30 degree angle, as shown
below.
30 degree
angle from
tangent
R
F
Answer: The force and lever arm are not
perpendicular, so we have to use the component of the
force that is perpendicular.  = (10 cos 30)(.5)
Use the right hand rule to determine the direction of torque.
When something simply rotates or spins
(like our wheel attached to a wall), it
isn’t going anywhere, so it has no linear
velocity.
Instead, we describe how fast it is
spinning with its angular velocity.
Watch the spinning girl at
http://www.theness.com/neurologicablog/?p=27
Angular velocity tells how far something
turns in a certain amount of time. The
amount it turns is an angle or angular
displacement.
Let’s consider the Earth:
(a) What is the magnitude of the
angular velocity (or angular
speed) of the Earth as it spins on
it axis?
(b) What is the direction of the
Earth’s angular velocity?
Answers:
As we look at this clock face:
(a) What is the angular velocity of the hour hand?
(b) What is the angular velocity of the minute hand?
(c) What is the angular velocity of the second hand?
(d) What is the direction of the torque the clock motor
applies to make these hands move?
The mass of an object helps us to
describe the amount of force it
will take it to start it in linear
motion or to stop it.
However, mass alone isn’t
enough to help us describe
rotational motion.
To describe how an object will
rotate….and the amount of torque
required to start it or stop it….we need to
know the mass of the object and how far
from the pivot the mass is concentrated.
We call this the moment of inertia or
rotational inertia of the object.
Moment of inertia is mass times the radius of
the object squared times a coefficient that
describes how far from the center the mass is
concentrated.
An object has that moment of inertia whether
it is spinning or stationary.
For example:
2
solid ball: I = 2/5 mr
hollow ball: I = 2/3 mr2
Objects that are spinning have angular momentum, L,
that depends upon the moment of inertia of the object
and how fast it is spinning (angular velocity):
L = I
Use the right hand rule to determine the direction
of angular velocity and angular momentum--which are in the same direction.
When this skater brings in his arms, he begins to spin faster.
He has decreased his radius of rotation, so his moment of
inertia is decreased.
By the law of conservation of angular momentum, a
decrease in moment of inertia means an increase in angular
speed. www.youtube.com [Search “one foot spin”]
Torque
R
F
Remember that torque, , is the product
of force and lever arm.
Another way of stating this is that
“torque is the applied force times the
perpendicular distance from the line of
force to the pivot”.
If the green board is balanced at its center on the
fulcrum and a force, F, is applied as shown, then the
distance L is the “lever arm”, since it is the
perpendicular distance from the line of the force to
the pivot or fulcrum.
F
L
The force produces a torque that is equal to F times L
and produces a clockwise rotation of the board.
Now, if a another force that is equal to the first is
applied at the same distance on the other side of the
pivot, it produces an equal amount of torque but in
the opposite direction.
F
L
The net torque now adds to zero—and the board does
not rotate. The board is in rotational equilibrium.
Note: This will only be true if the board is uniform and the pivot is at
the center of the board, so that the gravitational force is causing no
torque on the board.
Objects are in static equilibrium iff:
1. The net force in the x-direction is
zero.
2. The net force in the y-direction is
zero.
3. The net torque is zero.
Problems assigned in Chapter 8:
3, 23, 25
Problems assigned in Chapter 9:
3, 5, 9, 11,15, 20, 21, 26, 27