Dynamic Celt. Context: Product of inertia affects rotational motion

Transcription

1 Dnamic Celt From earliest recorded histor, children have been skipping stones and spinning tops. The challenge in phsics is to provide scientific eplanations for their curious behavior. Until now, the concept of product of inertia and its affect on rotational motion was difficult to demonstrate or understand. Designed for science instruction, the Dnamic Celt (also called a rattleback or wobblestone) provides visual, tactile, and auditor insights into the cause and affect of product of inertia. Eas to use - b setting a screw and placing it on a table, the Dnamic Celt can simpl spin, or spin and wobble, or reverse its spin! Created specificall for demonstration and laborator instruction b dnamics professors at Stanford, this is the first time the Dnamic Celt has been publicl available. 3 Beond its educational appeal, the Dnamic Celt is a friction-verification test for professional motion software that simulates spacecraft, aircraft, automobiles, machiner, and bio-devices. This rotational motion and product of inertia lab provides the three C s of scientific inquir: Concept: ( what is it ) Contet: ( wh is it important ) Calculation: ( how to form it ) Contet: Product of inertia affects rotational motion Product of inertia affects the rotation motion of objects. As seen in the net eperiments, the product of inertia has a profound affect on the spin, pitch, and roll rotational motions of a Dnamic Celt. 3 For information or to purchase a Dnamic Celt, visit Arbor Scientific and Search for Dnamic Celt. For video and more information, visit click on Video, and search for Dnamic Celt or visit Pages/ForceMotion/DnamicCelt.htm. Copright c b Paul Mitigu 1

2 Spin motions of a Dnamic Celt To investigate the affect of product of inertia on spin motions, ensure the Dnamic Celt looks like the one shown to the right (zero product of inertia). Place the Dnamic Celt on a flat, hard surface and spin it so it rotates clockwise. After observing its motion, spin it counterclockwise and circle the best description of the resulting motion. nitial motion Clockwise spin Simple spin Spin reversal and wobble Counter-clockwise spin Simple spin Spin reversal and wobble Net, adjust the bar so that it still has zero product of inertia but with the bar adjusted so it looks like the figure to the right. Place the Dnamic Celt on a flat, hard surface and spin it so it rotates clockwise. After observing its motion, spin it counter-clockwise. nitial motion Clockwise spin Simple spin Spin reversal and wobble Counter-clockwise spin Simple spin Spin reversal and wobble Now, adjust the bar so that it has a negative product of inertia (it should look something like the figure shown to the right). Place the Dnamic Celt on a flat, hard surface and spin it so it rotates clockwise. After observing its motion, spin it counter-clockwise. nitial motion Clockwise spin Simple spin Spin reversal and wobble Counter-clockwise spin Simple spin Spin reversal and wobble Net, adjust the bar so that it has a positive product of inertia (it should look something like the figure shown to the right). Spin the Dnamic Celt so it rotates clockwise. After observing its motion, spin it counter-clockwise. nitial motion Clockwise spin Simple spin Spin reversal and wobble Counter-clockwise spin Simple spin Spin reversal and wobble Adjust the bar so it makes a 5 angle with the Celt s long-ais. Spin the Celt counter-clockwise. Readjust the bar to an angle of 45 and repeat. Circle the angle associated with the resulting motion. Description of motion Slowl increasing wobble Long lasting wobble Smallest amplitude wobble Angle Copright c b Paul Mitigu 2

3 Pitch motions of a Dnamic Celt To investigate the affect of product of inertia on pitch motions, ensure the Dnamic Celt looks like the one shown to the right (zero product of inertia) and place it on a flat, hard surface. Touch its end so it rocks up and down. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Net, adjust the bar so that it still has zero product of inertia but with the bar adjusted so it looks like the figure to the right. Touch the end of the Dnamic Celt so it rocks up and down. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Now, adjust the bar so that it has a negative product of inertia. Touch its end so it rocks up and down. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Net, adjust the bar so that it has a positive product of inertia. Touch its end so it rocks up and down. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Summar of pitching motions Complete the following table which associates product of inertia with motion that results from an initial pitching motion (i.e., the Dnamic Celt is initiall set into motion b pushing down one of its ends). Product of inertia Zero Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Negative Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Positive Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Copright c b Paul Mitigu 3

4 Roll motions of a Dnamic Celt To investigate the affect of product of inertia on roll motions, ensure the Dnamic Celt looks like the one shown to the right (zero product of inertia) and place it on a flat, hard surface. Press its side so it strongl rolls from side to side. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Mostl roll Roll and weak clockwise spin Roll and weak counter-clockwise spin Now, adjust the bar so that it has a negative product of inertia. Press its side so it rolls from side to side. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Mostl roll Roll and weak clockwise spin Roll and weak counter-clockwise spin Net, adjust the bar so that it has a positive product of inertia. Press its side so it rolls from side to side. Mostl pitch Pitch and clockwise spin Pitch and counter-clockwise spin Mostl roll Roll and weak clockwise spin Roll and weak counter-clockwise spin Summar of spin, pitch, and roll preferences Consider a Dnamic Celt with a negative product of inertia. For each initial motion, circle the final spin direction. nitial motion Preferred spin direction Spin Clockwise Counter-clockwise Pitch Clockwise Counter-clockwise Roll Clockwise Counter-clockwise Consider a Dnamic Celt with a positive product of inertia. For each initial motion, circle the final spin direction. nitial motion Preferred spin direction Spin Clockwise Counter-clockwise Pitch Clockwise Counter-clockwise Roll Clockwise Counter-clockwise Copright c b Paul Mitigu 4

5 How it works? For more than a centur, the Dnamic Celt has attracted the attention of eminent phsicists. Most agree that its curious behavior results from three things: Friction at the point of contact between Celt and surface A curved surface with two different radii of curvature Principal aes of curvature that are not aligned with the principal aes of inertia (i.e., there is a non-zero product of inertia associated with the principal aes of curvature) Although detailed mathematical analses are able to accuratel predict the Celt s behavior, there is still great difficultl in understanding the mathematics in phsical terms. uoting Mont Hubbard (a prominent scientist who published detailed eperimental and theoretical work on the Celt) don t intuitivel understand it. f ou find the Dnamic Celt s motion perpleing, ou are in good compan. n terms of energ, phsicists eplain the Celt s behavior as energ echange between spin, pitching oscillations, and rolling oscillations. n terms of forces, several pla a role in the Celt s motion: 4 Normal and friction contact forces on the Dnamic Celt Local gravit forces Centrifugal forces and related forces due to acceleration Perhaps the best answer to wh does the Dnamic Celt move this is: F = m a 4 Humans seem to have a better intuitive understanding of the cause and affect of contact and distance forces (e.g., friction and gravit) than the centrifugal and Coriolis forces associated with acceleration. Copright c b Paul Mitigu 5

6 Concept of product of inertia: What is it? Product of inertia is a measure of the smmetr of mass distribution for two directions. To begin this investigation, guess at which of the following uniform-densit objects have a non-zero product of inertia (denoted ) and which have a zero product of inertia b completing each blank with = or. To make this determination, visuall sum the mass distribution in quadrants and and decide if that sum is equal to the mass distribution in quadrants and. f the mass distribution in quadrants and is equal to that in quadrants and, =0, otherwise 0. = 0 = 0 = 0 = 0 = 0 0 = 0 = 0 = = 0 Copright c b Paul Mitigu 6

7 Calculations: Product of inertia for a single particle The product of inertia of a single particle about a point for the ˆ and ŷ directions is calculated b the formula = -m where: m is the mass of is how far is from in the ˆ direction is how far is from in the ŷ direction. For eample, if m =1kg, =2m, and =3m, = -(1 kg) (2 m) (3 m) = -6 kgm 2 Referring to the figures below and knowing that particle has a mass m = 1 kg and that each tick-mark represents 1 m, calculate s product of inertia about point for each of the following. = -6 kgm 2 = 0 kg m 2 = 6 kg m 2 = 0 kg m 2 = -3 kgm 2 = 0 kg m 2 = 1 kg m 2 = 0 kg m 2 Circle the correct answer (negative, zero, or positive) for each statement about particle. When is in quadrant, its product of inertia is negative /zero/positive. When is in quadrant, its product of inertia is negative/zero/ positive. When is in quadrant, its product of inertia is negative /zero/positive. When is in quadrant, its product of inertia is negative/zero/ positive. When is on a quadrant boundar, its product of inertia is negative/ zero /positive. Copright c b Paul Mitigu 7

8 Calculations: Product of inertia for a sstem of particles The product of inertia of a sstem of particles is simpl the sum of the products of inertias of each of the individual particles. For eample, the product of inertia of particles 1 and 2 about point for the ˆ and ŷ directions is calculated b the formula 2 ( 2, 2 ) = -m m where: m i is the mass of particle i (i=1, 2) i is how far i is from in the ˆ direction. i is how far i is from in the ŷ direction. ( 1, 1 ) For eample, if m 1 =1kg, 1 =3m, 1 =1m, and m 2 =2kg, 2 =2m, 2 =3m, = -(1kg)(3m)(1m) + -(2kg)(2m)(3m) = -15 kg m 2 Referring to the figures below and knowing that each particle has a mass of 1 kg and that each tick-mark represents 1 m, calculate the sstem s product of inertia about point for each of the following = -12 kg m 2 = 0 kg m 2 = 12 kg m 2 = 0 kg m 2 Circle the correct answer (negative, zero, or positive) for each of the following statements. When the particles are in quadrants and, product of inertia is negative /zero/positive. When the particles are in quadrants and, product of inertia is negative/zero/ positive. When the particles are on quadrant boundaries, product of inertia is negative/ zero /positive. Copright c b Paul Mitigu 8