Chapter 11Chapter 11

Rotational MechanicsRotational Mechanics

Rotational InertiaRotational Inertia An object rotating about an axis tends to An object rotating about an axis tends to

remain rotating unless interfered with by remain rotating unless interfered with by some external influence.some external influence.

This influence (rotation) is called This influence (rotation) is called torquetorque.. Torque is Torque is notnot a force. (Torque causes rotation a force. (Torque causes rotation

and Force causes acceleration.)and Force causes acceleration.) Torque Torque isis applied leverage. applied leverage.

TorqueTorque Torque is the product of the force and lever-Torque is the product of the force and lever-

arm distance, which tends to produce arm distance, which tends to produce rotation.rotation.

Torque = force Torque = force lever arm lever arm» wrencheswrenches

Balanced Torque:Balanced Torque:– F x F x short lever armshort lever arm = = FF x long lever arm x long lever arm– see-saws (Figure 11.5) or scale balances with see-saws (Figure 11.5) or scale balances with

sliding weightssliding weights

Balanced TorqueBalanced Torque

SeeSaw - Figure 11.5SeeSaw - Figure 11.5– Force x distance on each side of seesaw has to Force x distance on each side of seesaw has to

be equal. be equal. – What if the boy is 600N, how far would he What if the boy is 600N, how far would he

have to sit from the fulcrum for equilibrium?have to sit from the fulcrum for equilibrium? 1m

TorqueTorque

Torque Feeler LabTorque Feeler Lab Other : “twisting forces”Other : “twisting forces”

– Prying lid off a can with screwdriverPrying lid off a can with screwdriver– Turning a wrenchTurning a wrench– Opening a doorOpening a door– Steering wheel (modified wrench)Steering wheel (modified wrench)

Consider 1) the application of a Consider 1) the application of a force and 2) leverageforce and 2) leverage

Weighing an Elephant LabWeighing an Elephant Lab (F (Fd)d)AA = (F = (Fd)d)BB

A meter stick is suspended at its midpoint A meter stick is suspended at its midpoint and two blocks are attached along its and two blocks are attached along its length. A 10-N block is attached 20cm to length. A 10-N block is attached 20cm to the left of the midpoint. Where must a 40-the left of the midpoint. Where must a 40-N block be placed in order to keep the N block be placed in order to keep the meter stick balanced? Show your meter stick balanced? Show your calculationscalculations

Torque ProblemsTorque Problems

In the previous question, if the 10-N block was changed In the previous question, if the 10-N block was changed to a 80-N block, how does the location of the 40-N to a 80-N block, how does the location of the 40-N block change?block change?

To remove a nut from an old rusty bolt, you apply a To remove a nut from an old rusty bolt, you apply a 100-N force to the end of a wrench perpendicular to the 100-N force to the end of a wrench perpendicular to the wrench handle. The distance from the applied force to wrench handle. The distance from the applied force to the axis of the bolt is 25 cm. What is the torque exerted the axis of the bolt is 25 cm. What is the torque exerted on the bolt in Non the bolt in N..m?m?

Rotational Inertia Rotational Inertia http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechttp://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/hanics/circularmotion/hewitt/Source_Files/08_RotationalInertiaHam_VID.mov08_RotationalInertiaHam_VID.mov

https://www.youtube.com/watch?v=CHQOctEvtTYhttps://www.youtube.com/watch?v=CHQOctEvtTY

Rotational InertiaRotational InertiaThe greater the rotational inertia, the more difficult it is to The greater the rotational inertia, the more difficult it is to

change the rotational speed of an object.change the rotational speed of an object. The resistance of an object to change in its rotational The resistance of an object to change in its rotational

motion is called rotational inertia (motion is called rotational inertia (moment of inertia)moment of inertia).. A torque is required to change the rotational state of A torque is required to change the rotational state of

motion of an object.motion of an object. Rotational inertia depends on mass and how the mass is Rotational inertia depends on mass and how the mass is

distributed. distributed. The greater the distance between the The greater the distance between the object’s mass concentration and the axis of rotation, the object’s mass concentration and the axis of rotation, the greater the rotational inertia.greater the rotational inertia.– A short pendulum has less rotation inertia and therefore A short pendulum has less rotation inertia and therefore

swings back and forth more frequently than a long pendulum. swings back and forth more frequently than a long pendulum. – Bent legs swing back and forth more easily than outstretched Bent legs swing back and forth more easily than outstretched

legs.legs.

Law of Rotational InertiaLaw of Rotational InertiaAn object rotating about an axis tends to keep rotating An object rotating about an axis tends to keep rotating

about that axis in absence of an external torque.about that axis in absence of an external torque. Rotational Inertia (Moment of Inertia) is the resistance Rotational Inertia (Moment of Inertia) is the resistance

of an object to change its rotational motion.of an object to change its rotational motion. The greater the distance of mass concentration, the The greater the distance of mass concentration, the

greater the resistance to rotation (rotational inertia)greater the resistance to rotation (rotational inertia)– Balance a weight on finger Balance a weight on finger – Balance a weight on long stick (similar to balancing a broom Balance a weight on long stick (similar to balancing a broom

or long handled hammer)or long handled hammer)– Long legged peoples’ gaits to short peoples’ gaitLong legged peoples’ gaits to short peoples’ gait– Similar to adjustments needed to keep rocket vertical when Similar to adjustments needed to keep rocket vertical when

first fired.first fired.– Tightrope walkers carry long polesTightrope walkers carry long poles– Training wheels on beginner bicycleTraining wheels on beginner bicycle

Formulas for Rotational InertiaFormulas for Rotational Inertia

Fig. 11.14 pg.157Fig. 11.14 pg.157 Don’t memorize them. Mass more spread Don’t memorize them. Mass more spread

out the rotational inertia (I) is less.out the rotational inertia (I) is less.

Rotational Inertia & RollingRotational Inertia & Rollinghttp://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_WhyABallRollsDo_VID.movhttp://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 Objects of the same shape but different sizes accelerate equally when rolled down an incline.equally when rolled down an incline.

An object with a greater rotational inertia takes more An object with a greater rotational inertia takes more time to get rolling than an object with a smaller time to get rolling than an object with a smaller rotational inertia. rotational inertia. – A hollow cylinder rolls down an incline much slower than a A hollow cylinder rolls down an incline much slower than a

solid cylinder.solid cylinder. Shapes with greater rotational inertia (“laziness”) lag Shapes with greater rotational inertia (“laziness”) lag

behind shapes with less rotational inertia. Greater behind shapes with less rotational inertia. Greater rotational inertia is the one with its mass concentrated rotational inertia is the one with its mass concentrated farther from axis of rotation.farther from axis of rotation.

All objects of the same shape roll down an incline with All objects of the same shape roll down an incline with the same acceleration, even if their masses are different.the same acceleration, even if their masses are different.

Rotational Inertia & GymnasticsRotational Inertia & Gymnastics

The three principal axes of rotation in the human body are the The three principal axes of rotation in the human body are the longitudinal axis, the tranverse axis and the medial axis.longitudinal axis, the tranverse axis and the medial axis.

The three axes of rotation in the human body are at right The three axes of rotation in the human body are at right angles to one another. angles to one another.

All three axes pass through the CG of the body.All three axes pass through the CG of the body. Vertical axis that passes from head to toe is the Vertical axis that passes from head to toe is the longitudinal longitudinal

axisaxis. Rotational inertia about this axis is increased by . Rotational inertia about this axis is increased by extending a leg or the arms.extending a leg or the arms.

Somersault or flip rotates you around your Somersault or flip rotates you around your tranverse axistranverse axis. . Tucking in your legs and arms reduces your rotational Tucking in your legs and arms reduces your rotational inertia; straighten legs & arms to increase rotational inertia inertia; straighten legs & arms to increase rotational inertia about this axisabout this axis

Medial axisMedial axis is front-to-back axis. You rotate is front-to-back axis. You rotate about the medial axis when you do a cartwheel.about the medial axis when you do a cartwheel.

Rotational Inertia & GymnasticsRotational Inertia & GymnasticsJust as the body can change shape and Just as the body can change shape and

orientation, the rotational inertia of the body orientation, the rotational inertia of the body changes also.changes also.

Read pg. 159-160 togetherRead pg. 159-160 together. . Notice the rotational inertia about any of the Notice the rotational inertia about any of the

axes does axes does notnot depend on direction of spin. depend on direction of spin. Rotational inertia is different for the same Rotational inertia is different for the same

body configuration about different axes.body configuration about different axes. US NSF - News - Science of the Olympic Winter Games - Figuring Out Figure SkatingUS 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)US NSF - News - Science of the Olympic Winter Games - Aerial Physics (Aerial Skiing)

Angular MomentumAngular MomentumNewton’s First Law of Inertia for rotating systems states that Newton’s First Law of Inertia for rotating systems states that

an object or system of objects will maintain its angular an object or system of objects will maintain its angular momentum unless acted upon by an unbalanced torque.momentum unless acted upon by an unbalanced torque.

http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_ConservationOfAng_VID.movhttp://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_ConservationOfAng_VID.mov

Linear momentum = mass x velocityLinear momentum = mass x velocity angular momentum = rotational inertia angular momentum = rotational inertia rotational velocity rotational velocity

L = I L = I When a direction is assigned to rotational speed, it is called When a direction is assigned to rotational speed, it is called

rotational velocityrotational velocity.. Angular momentum is a Angular momentum is a vector quantityvector quantity and and has directionhas direction as as

well as well as magnitudemagnitude..

II II

Angular MomentumAngular Momentum When an object is small compared with the radial When an object is small compared with the radial

distance to its axis of rotation, its angular momentum is distance to its axis of rotation, its angular momentum is equal to the magnitude of its linear momentum, mequal to the magnitude of its linear momentum, mvv, , multiplied by the radial distance, r.multiplied by the radial distance, r.

angular momentum = mangular momentum = mvvrr– A moving bicycle is easier to balance than a bicycle at rest because A moving bicycle is easier to balance than a bicycle at rest because

of the angular momentum provided by the spinning wheels.of the angular momentum provided by the spinning wheels.

If object is small compared to the radius:If object is small compared to the radius:

– I = mvrI = mvr

Conservation of Angular MomentumConservation of Angular Momentum Angular momentum is conserved when no external Angular momentum is conserved when no external

torque acts on an object\torque acts on an object\ A person who spins with arms extended obtains A person who spins with arms extended obtains

greater rotational speed when arms are drawn in.greater rotational speed when arms are drawn in. Zero-angular-momentum twists & turns can be Zero-angular-momentum twists & turns can be

performed by turning one part of the body against the performed by turning one part of the body against the other.other.

Examples: Examples: – 1. ice skater spin1. ice skater spin

– 2. cat dropped on back2. cat dropped on back

– 3. diving3. diving