Transcript Rotational Inertia

Chapter 11
Rotational Mechanics
Rotational Inertia
An object rotating about an axis tends to
remain rotating unless interfered with by
some external influence.
 This influence (rotation) is called torque.
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Torque is not a force. (Torque causes rotation
and Force causes acceleration.)
Torque is applied leverage.
Torque
Torque is the product of the force and leverarm distance, which tends to produce
rotation.
 Torque = force   lever arm
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» wrenches
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Balanced Torque:
– F x short lever arm = F x long lever arm
– see-saws (Figure 11.5) or scale balances with
sliding weights
Balanced Torque
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SeeSaw - Figure 11.5
– Force x distance on each side of seesaw has to
be equal.
– What if the boy is 600N, how far would he
have to sit from the fulcrum for equilibrium?
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1m
Torque
Torque Feeler Lab
 Other : “twisting forces”
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Prying lid off a can with screwdriver
Turning a wrench
Opening a door
Steering wheel (modified wrench)
Consider 1) the application of a
force and 2) leverage
Weighing an Elephant Lab
(Fd)A = (Fd)B
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A meter stick is suspended at its midpoint
and two blocks are attached along its length.
A 10-N block is attached 20cm to the left of
the midpoint. Where must a 40-N block be
placed in order to keep the meter stick
balanced? Show your calculations
Torque Problems
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In the previous question, if the 10-N block was changed
to a 80-N block, how does the location of the 40-N
block change?
To remove a nut from an old rusty bolt, you apply a
100-N force to the end of a wrench perpendicular to the
wrench handle. The distance from the applied force to
the axis of the bolt is 25 cm. What is the torque exerted
on the bolt in N.m?
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Rotational Inertia
http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mec
hanics/circularmotion/hewitt/Source_Files/08_RotationalIn
ertiaHam_VID.mov
https://www.youtube.com/watch?v=CHQOctEvtTY
Rotational Inertia
The greater the rotational inertia, the more difficult it is to
change the rotational speed of an object.
 The resistance of an object to change in its rotational
motion is called rotational inertia (moment of inertia).
 A torque is required to change the rotational state of
motion of an object.
 Rotational inertia depends on mass and how the mass is
distributed. The greater the distance between the
object’s mass concentration and the axis of rotation, the
greater the rotational inertia.
– A short pendulum has less rotation inertia and therefore
swings back and forth more frequently than a long pendulum.
– Bent legs swing back and forth more easily than outstretched
legs.
Law of Rotational Inertia
An object rotating about an axis tends to keep rotating
about that axis in absence of an external torque.
 Rotational Inertia (Moment of Inertia) is the resistance
of an object to change its rotational motion.
 The greater the distance of mass concentration, the
greater the resistance to rotation (rotational inertia)
– Balance a weight on finger
– Balance a weight on long stick (similar to balancing a broom
or long handled hammer)
– Long legged peoples’ gaits to short peoples’ gait
– Similar to adjustments needed to keep rocket vertical when
first fired.
– Tightrope walkers carry long poles
– Training wheels on beginner bicycle
Formulas for Rotational Inertia
Fig. 11.14 pg.157
 Don’t memorize them. Mass more spread
out the rotational inertia (I) is less.
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Rotational Inertia & Rolling
http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_WhyABallRollsDo_VID.mov
Objects of the same shape but different sizes accelerate
equally when rolled down an incline.
 An object with a greater rotational inertia takes more
time to get rolling than an object with a smaller
rotational inertia.
– A hollow cylinder rolls down an incline much slower than a
solid cylinder.
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Shapes with greater rotational inertia (“laziness”) lag
behind shapes with less rotational inertia. Greater
rotational inertia is the one with its mass concentrated
farther from axis of rotation.
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All objects of the same shape roll down an incline with
the same acceleration, even if their masses are different.
Rotational Inertia & Gymnastics
The three principal axes of rotation in the human body are the
longitudinal axis, the tranverse axis and the medial axis.
 The three axes of rotation in the human body are at right
angles to one another.
 All three axes pass through the CG of the body.
 Vertical axis that passes from head to toe is the longitudinal
axis. Rotational inertia about this axis is increased by
extending a leg or the arms.
 Somersault or flip rotates you around your tranverse axis.
Tucking in your legs and arms reduces your rotational
inertia; straighten legs & arms to increase rotational inertia
about this axis
 Medial axis is front-to-back axis. You rotate
 about the medial axis when you do a cartwheel.
Rotational Inertia & Gymnastics
Just as the body can change shape and
orientation, the rotational inertia of the body
changes also.
Read pg. 159-160 together.
 Notice the rotational inertia about any of the
axes does not depend on direction of spin.
 Rotational inertia is different for the same
body configuration about different axes.
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US NSF - News - Science of the Olympic Winter Games - Figuring Out Figure Skating
US NSF - News - Science of the Olympic Winter Games - Aerial Physics (Aerial
Skiing)
Angular Momentum
Newton’s First Law of Inertia for rotating systems states that
an object or system of objects will maintain its angular
momentum unless acted upon by an unbalanced torque.
http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_ConservationOfAng_VID.mov
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Linear momentum = mass x velocity
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angular momentum = rotational inertia  rotational velocity
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L=Iw
When a direction is assigned to rotational speed, it is called
rotational velocity.
Angular momentum is a vector quantity and has direction as
well as magnitude.
I w = I w
Angular Momentum
 When an object is small compared with
the radial distance to its axis of rotation,
its angular momentum is equal to the
magnitude of its linear momentum, mv,
multiplied by the radial distance, r.
angular momentum = mvr
– A moving bicycle is easier to balance than a
bicycle at rest because of the angular momentum
provided by the spinning wheels.
Conservation of Angular Momentum
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Angular momentum is conserved when no external
torque acts on an object\
A person who spins with arms extended obtains
greater rotational speed when arms are drawn in.
Zero-angular-momentum twists & turns can be
performed by turning one part of the body against the
other.
Examples:
– 1. ice skater spin
– 2. cat dropped on back
– 3. diving