Download rotational inertia

Rotational Mechanics
Center of Gravity
The point located
at the object’s
average position
of weight.
The center of gravity does not always
lie within the object itself….
When an object is in motion, its center of
mass will follow a smooth line.
Center of Mass and Center of
Gravity
• Toss a baseball into the air, and it will follow a
smooth parabolic trajectory.
• Toss a baseball bat spinning into the air, and its
path is not smooth; its motion is wobbly, and it
seems to wobble all over the place.
Locating the Center of Gravity
Method 1
• Balancing an object provides a simple
method of locating its center of gravity.
• The weight of an entire stick behaves as if
it were concentrated at the stick’s center of
gravity.
Locating the Center of Gravity, CG
Method Two:
If you suspend any object, its cg will be
located along a vertical line drawn from
the suspension point.
• For a shape that is irregular, the center of gravity lies
directly beneath the point of suspension. If we draw a
vertical line through the point of suspension, the center
of gravity is somewhere along that line.
• To determine exactly where it lies, we have to suspend
the object from some other point and draw a vertical line
from that point of suspension.
• Where the two lines intersect is the center of gravity.
If the cg remains in a vertical line with the
“support base”, an object will not topple
over.
However, if the cg is out beyond the support
base, it will topple.
Why do pregnant women get backaches?
Pre-AP: Finding the coordinate for
the center of mass for 2 separate
masses.
xcm
m1 x1  m2 x2

m1  m2
When there’s no net external forces acting
on the masses, if the center of mass was
at rest, it will remain at rest, if it was in
motion, it continues that motion.
xcm
m1 x1  m2 x2

m1  m2
m2
m1
x1
x2
Get out your map of Texas
to turn in.
Put your name on the back.
• Consider the force
required to open a door.
Is it easier to open the
door by pushing/pulling
away from hinge (by the
door knob) or close to
hinge?
Farther from
from hinge=
larger
rotational
effect!
close to hinge
away from
hinge
Torque
• Torque, , is the tendency of a force to
rotate an object about some axis
Door example:
  Fd
Unit: Newton x meter, Nm
–  is the torque
– d is the lever arm
– F is the force
In the illustration below, more torque is applied in
the middle example than in the first example
because the Force is more effective when it is
perpendicular.
In the third example a piece of pipe is used to
extend the lever arm providing even more
torque.
ConcepTest
You are using a wrench
and trying to loosen a
rusty nut.
Which of the
arrangements shown is
most effective in
loosening the nut?
List in order of
descending efficiency.
Answer: 2, 1 & 4, 3
Pre-AP Lever Arm
In this case, the lever arm is NOT L.
The lever arm is d, the shortest
perpendicular distance from the axis of
rotation (the nut) to a line drawn along the
direction of the force
Lever arm = d = L sin Φ
Torque = Force x lever arm
Torque = FLsin Φ
The lever arm is not necessarily the distance
between the axis of rotation and point
where the force is applied
An Alternative Look at Torque
• The force could also be
resolved into its x- and ycomponents
– The x-component, F cos Φ,
produces 0 torque
– The y-component, F sin Φ,
does produce torque:
  FL sin 
F is the force
L is the distance along the object
Φ is the angle between force and object
L
What if two or more different forces
act along a lever arm?
Net Torque
• The net torque is the
sum of all the torques
Torque = Force x lever arm
produced by all the
forces
lever arm lever arm
Force = weight
Force = weight
Net Torque = Weight 1 x lever arm 1 – Weight 2 x lever arm 2
Balanced Torque occurs when the net torque is ZERO
The torque on one side of the fulcrum = the torque on the other side
Where would the 300 N boy have to sit
relative to fulcrum for balanced
torque?
3 meters
??
300 N
200 N
200 x 3 = 300 x ??
• Notice how the torque is
the same for the boy as it
is for the girl.
• Even if the girl is
suspended by a rope 4
feet below where she was
the torques are still the
same!
• The lever arm is the
perpendicular distance
from the pivot point to the
line along which the force
acts.
weight
QUESTION
• A uniform meterstick supported at the 25cm mark balances when a 2-kg rock is
suspended at the 0-cm end. What is the
mass of the meterstick?
25 cm
2 kg
25 cm
What is the location where all the weight
of the meter stick is acting?
Weight
of meter
stick
?
ANSWER
The mass of the meterstick is 2 kg.
The system is in equilibrium, so any torques
must be balanced.
Review
• Torque is the rotational counterpart of force.
• Force tends to change the motion of things;
• Torque tends to twist or change the state of
rotating things.
For torque, the perpendicular distance from the
axis of rotation is called the lever arm.
Torque is the product of the lever arm and the
force that tends to produce rotation:
Torque = lever arm x Force
Angular Momentum
Rotational Inertia
Newton’s 1st Law, the Law of Inertia
“An object in motion tends to remain in
motion, unless a net force acts on it.”
Its tendency to either remain at rest or in
motion is called its inertia and is
measured by taking its mass.
In a similar way,
“An object in rotation tends to keep rotating”
The resistance of an object to changes in its
rotation is its rotational inertia.
Rotational inertia, I , depends on mass and how
that mass is distributed around the axis of
rotation. The greater the distance between the
bulk of the mass and the axis of rotation, the
greater the rotational inertia.
Hoop: the bulk of the mass is as far
away from the axis as it could
possibly be!
Maximum rotational inertia!
This means this shape is the hardest
to start rotating AND the hardest to
stop rotating.
Rotational inertia, I , depends on mass and how
that mass is distributed around the axis of
rotation. The greater the distance between the
bulk of the mass and the axis of rotation, the
greater the rotational inertia.
Spherical shell: the bulk of the mass
is as far away from the axis as it
could possibly be!
Lots of rotational inertia (but not as
much as a hoop)!
This means this shape is the hard to
start rotating AND the hard to stop
rotating.
Rotational inertia, I , depends on mass and how
that mass is distributed around the axis of
rotation. The greater the distance between the
bulk of the mass and the axis of rotation, the
greater the rotational inertia.
cylinder: the bulk of the mass is
evenly distributed
Not as much rotational inertia
This means this shape is the easier
to start rotating AND the easier to
stop rotating.
Rotational inertia, I , depends on mass and how
that mass is distributed around the axis of
rotation. The greater the distance between the
bulk of the mass and the axis of rotation, the
greater the rotational inertia.
Solid sphere: the mass is as closely
packed as possible!
Least rotational inertia!
This means this shape is the easiest
to start rotating AND the easiest to
stop rotating.
A person can change their own rotational inertia by
extending your arms, rolling up in a ball, etc.
By doing that, you will rotate or spin more easily or
less easily, depending on if you increased or
decreased your rotational inertia.
Gymnasts, ice-skaters, divers, and dancers all use
this principle.
Objects that are
rotating have more
STABILITY!
Think of frisbees and
bicycles.
Angular Momentum
Object in rotational motion have rotational or
angular momentum, L, found by multiplying
rotational inertia x angular velocity
I - rotational inertia
w (omega)- angular (rotational) velocity
measured in radians per second.
Angular Momentum, L = I w
Since angular momentum, Iw, remains constant, if
your rotational inertia, I, goes down…
Your velocity, w, goes up!
But the product of inertia and velocity (original and
final) stays the same!
Iowo = Ifwf
Examples:
• Kids on a merry-go-round
• Ice-skater
• Diver
• Student on a spinning chair
Example: An ice skater with a rotational
inertia of 80 kg m2 is spinning with velocity
of 2 rad/s. When she pulls her arms in, her
rotational inertia drops to 5 kg m2. What is
her angular momentum? What is her final
velocity?
Io= 80
wo = 2
If  5
wf = ?
Angular momentum, L = Iowo = 80 x 2 = 160
Original L =
Final L
Iowo = Ifwf
wf= 80(2) / 5
wf= 32 rad / s
Pre-AP Don’t forget….
v = wr
w = v/r