r some or all of m, x, yz. Similarly for I yy and I zz.

Transcription

1 Homework 10. Chapters 1, 1. Moments and products of inertia Concepts: What objects have a moment of inertia? (Section 1.1). Consider the moment of inertia S/ bu bu of an object S about a point for the unit vector û. n general, for S/ bu bu to be a positive real number, S should be a (circle all appropriate objects): Real number Matri Set of points Mass center of a rigid bod Vector Point Reference frame Fleible bod D orthogonal unit basis Particle Rigid bod Sstem of particles and bodies 10. Formulas for a particle s moments and products of inertia (Sections 1.1 and 1.). The figure shows a particle of mass m and right-handed orthogonal unit vectors n, n, n z. s position vector from a point is n + n + z n z. (m) n Epress ( s moment of inertia about for n ) in terms of r some or all of m,, z. Similarl for and. Epress ( s product of inertia about for n and n ) in z terms of some or all of m,, z. Similarl for z and z. n z Result: = ( + ) = = n = z = z = Circa 189, Gibbs invented the inertia dadic as a convenient suitcase for holding moments and products of inertia. Write s inertia dadic about in terms of n, n, n z and ij (i, j =,, z). f needed, refer to Section 1.1. = n n + n n + n n z + n n + n n Parallel ais theorem and moments of inertia (Section 1.1.). The sstem S shown to the right consists of particles 1 and, each of mass m, in a plane perpendicular to the unit vector n z. G n The shift theorem (also called the parallel ais theorem) shifts S s moment of inertia about S cm (the mass center of S) for the unit vector n z to an arbitrar point P in the plane using A n z C D F E n S/P = S/Scm + m S d where S/Scm is the sstem s moment of inertia about S cm for n z, m S is the mass of S, andd is the distance from S cm to P. Use the shift theorem to estimate the order of S s moment of inertia for the lines parallel to n z that pass through points A,, C, D, E, F, G, andh, respectivel. Note: Grid lines are equall spaced. Result: Smallest Largest Knowing each particle has mass m = 1 kg and the grid lines are spaced 1 m apart, calculate S s moment of inertia about A,, C, D, E, F, G, andh, respectivel. Result: S/A S/ (in kg m ) 8 S/C S/D S/E S/F S/G H S/H Copright c Paul Mitigu. All rights reserved. Homework 10: Moments/products of inertia

2 10. Sign conventions for products of inertia (Section 1..). There are two sign conventions ( ± ) for products of inertia which often lead to errors True/False. 10. Calculations: Product of inertia for a single particle (Section 1.). The product of inertia of a single particle about a point for the and ŷ directions is calculated b the formula = m m is the mass of is the measure of s position vector from is the ŷ measure of s position vector from For eample, if m =1kg, =m, and =m, = (1kg)(m)(m) = kgm Knowing particle has a mass of 1 kg and each tick-mark represents 1 m, calculate s product of inertia about point for each figure below. = kgm = kg m = kg m = kg m = kgm = kg m = kg m = kg m Circle the correct answer (negative, zero, or positive) for each statement about particle. When is in quadrant, is negative/zero/positive. When is in quadrant, is negative/zero/positive. When is in quadrant, is negative/zero/positive. When is in quadrant, is negative/zero/positive. When is on a quadrant boundar, is negative/zero/positive. Copright c Paul Mitigu. All rights reserved. 8 Homework 10: Moments/products of inertia

3 10. Calculations: Product of inertia for a sstem of particles (Section 1.). 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 about point for the and ŷ directions is calculated b the formula = i=1 mi i i m i is the mass of particle i (i=1, ) i is the measure of i s position vector from i is the ŷ measure of i s position vector from (, ) For eample, if m 1 =1kg, 1 =m, 1 =1m, and m =kg, =m, =m, 1 ( 1, 1 ) = m m = (1kg)(m)(1m) + (kg)(m)(m) = 1 kg m Knowing each particle has a mass of 1 kg and each tick-mark represents 1 m, calculate the sstem s product of inertia about point for each figure below = kg m = kg m = kg m 1 = kg m Circle the correct answer (negative, zero, or positive) for each of the following statements. When the particles are in quadrants and, is negative/zero/positive. When the particles are in quadrants and, is negative/zero/positive. When the particles are on quadrant boundaries, is negative/zero/positive. 10. Concepts: Products of inertia of propellers (Section 1..). The following shows four uniform-densit objects. For each object, consider the product of inertia of the object for lines that pass through point and are parallel to n and n. For each object, visuall determine whether the product of inertia is negative, zero, or positive. n z n n zero negative negative zero n Guess n z n Copright c Paul Mitigu. All rights reserved. 9 Homework 10: Moments/products of inertia

4 10.8 Concepts: Products of inertia what is? (Section 1..). Product of inertia is a measure of the smmetr of mass distribution in two directions about a point. To investigate this concept, use our geometrical insights (not equations) to determine which of the following uniform-densit objects have a negative, zero, or positive product of inertia. Visuall sum the mass distribution for in quadrants and and compare that to the mass distribution in quadrants and. Complete each blank below with < or = or >. f mass distribution in quadrants + is greater than in quadrants +, > 0. f mass distribution in quadrants + is smaller than in quadrants +, < 0. f mass distribution in quadrants + is equal to quadrants +, =0. = Purchase rattleback at Eplained: Copright c Paul Mitigu. All rights reserved. 0 Homework 10: Moments/products of inertia

5 10.9 Conceptual understanding of moments and products of inertia (Sections 1.1. and 1..). bjects A,, C, D, ande are all flat planar objects with uniform densit and the same mass. The circle and semi-circle s diameter, square and rectangle s width, and thin rod s length are equal. A C D E n S denotes one of n n z A,, C, D, ore. % Consider S/, S s moment of inertia about the line passing through point and parallel to n z. Knowing moment of inertia is mass distance, use visual estimates to list the objects in ascending order of S/. ftwoobjectshavethesamevalueof S/, group them together. Result: Smallest Largest 8% Consider S/Scm, S s moment of inertia about the line passing through S cm (the mass center of S) and parallel to n z. Use visual estimates to list the objects in ascending order of S/Scm. Note: A and E have nearl equal S/Scm. The tetbook s inertia appendi helps resolve their difference. Result: Smallest D Largest (given) % Consider S/, S s product of inertia for point and unit vectors n and n. For each object, visuall determine if S/ is negative ( ), zero (0), or positive (+). A C D E Result: Assembling inertia dadics from the tetbook s inertia appendi (Sections 1.1 and 1.). Referring to the following figures and the tetbook s inertia appendi, assemble / cm ( s inertia dadic about its center of mass cm) for the unit vectors b, b,. Epress results in matri form. b b b cm b bz b b b a 1 ma a 0 b a c m(a + b ) b r b cm b b r b h = b 1 b Copright c Paul Mitigu. All rights reserved. 1 Homework 10: Moments/products of inertia

6 10.11 iomechanics: Estimating mass distribution properties of a bone (Sections 1.1 and 1.). The following figure shows a relativel thin, uniform-densit, rigid bone. Right-handed orthogonal unit vectors b, b, are fied in. Apoint o of is midwa though the bone ( r cm/o =0). Estimate and draw the location of cm ( s center of mass) on the figure. b cm Hint: Draw horizontal (parallel to b ) and vertical (parallel to b ) lines passing through o. Similarl for cm. o b 0 cm Knowingthe1kgbonefitssnuglina0cmbcmb1cmbo,visuall estimate the following values. Note: ther than 0.0 which is used times, use each of the following values once in our table Description Smbol Approimate value b measure of cm s position vector from o. cm b measure of cm s position vector from o 1. cm s moment of inertia about cm for b s moment of inertia about cm for b s moment of inertia about cm for s product of inertia about cm for b and b s product of inertia about cm for b and s product of inertia about cm for b and s moment of inertia about o for b s moment of inertia about o for b s moment of inertia about o for /cm kg cm /cm kg cm /cm kg cm /cm kg cm /cm z kg cm /cm z kg cm /o kg cm /o kg cm /o kg cm s product of inertia about o for b and b s product of inertia about o for b and /o kg cm /o z kg cm s product of inertia about o for b and /o z kg cm Hint: Start with products of inertia, then smallest moments of inertia, then largest moments of inertia. Hint: Draw an end-view of the bone to help estimate the two z. Draw a top-view to estimate the two z Dadics and dot-products with orthogonal unit vectors b, b, (Section 1.1). Non-smmetric dadic: Smmetric dadic: N = b b + b b + b bz + bz b S = b b + b b + b bz + b + bz Non-smmetric dadic Smmetric dadic N ( b + b ) = b + b S ( b + b ) = + + ( b + b ) N = + + ( b + b ) S = + + N anvector = anvector N True/False S anvector = anvector S b True/False Copright c Paul Mitigu. All rights reserved. Homework 10: Moments/products of inertia