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Question-3337




Question Number 3337 by Yozzi last updated on 11/Dec/15
         ∗   ∗   ∗              ∗   ∗   ∗       ∗   ∗   ∗   ∗   ∗   ∗   ∗   ∗       ∗   ∗   ∗   ∗   ∗   ∗   ∗   ∗       ∗   ∗   ∗   ∗   ∗   ∗   ∗   ∗       ∗   ∗   ∗              ∗   ∗   ∗       ⇠⇠             ⇠⇠             h_1          h_2          h_1                    Figure 1  Let us model a yoyo by three combined, uniform  cylindrical bodies which yield the above profile  in Fig. 1. The mass of the entire body  is M.  h_1  is the side length of the two outter  cylindrical bodies and r_1  is the side radius of   these cylinders. h_(2 ) is the side length  of the inner cylindrical body whose side  radius is r_2 . Here, r_1 >r_2  and h_1 >h_2 .  The axis of each cylindrical body,  passing through their respective  centre of mass and perpendicular to  their circular faces, all coincide with  each other. Determine the 3D moment  of inertia of the yoyo about its axis  normal to its outter circular faces and passing  through its centre of mass.
$$\:\: \\ $$$$\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\:\:\:\:\:\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast \\ $$$$\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast \\ $$$$\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast \\ $$$$\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\ast \\ $$$$\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\:\:\:\:\:\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast \\ $$$$\:\:\:\:\:\dashleftarrow\dashleftarrow\:\:\:\:\:\:\:\:\:\:\:\:\:\dashleftarrow\dashleftarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{h}_{\mathrm{1}} \:\:\:\:\:\:\:\:\:{h}_{\mathrm{2}} \:\:\:\:\:\:\:\:\:{h}_{\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Figure}}\:\mathrm{1} \\ $$$${Let}\:{us}\:{model}\:{a}\:{yoyo}\:{by}\:{three}\:{combined},\:{uniform} \\ $$$${cylindrical}\:{bodies}\:{which}\:{yield}\:{the}\:{above}\:{profile} \\ $$$${in}\:{Fig}.\:\mathrm{1}.\:{The}\:{mass}\:{of}\:{the}\:{entire}\:{body} \\ $$$${is}\:{M}. \\ $$$${h}_{\mathrm{1}} \:{is}\:{the}\:{side}\:{length}\:{of}\:{the}\:{two}\:{outter} \\ $$$${cylindrical}\:{bodies}\:{and}\:{r}_{\mathrm{1}} \:{is}\:{the}\:{side}\:{radius}\:{of}\: \\ $$$${these}\:{cylinders}.\:{h}_{\mathrm{2}\:} {is}\:{the}\:{side}\:{length} \\ $$$${of}\:{the}\:{inner}\:{cylindrical}\:{body}\:{whose}\:{side} \\ $$$${radius}\:{is}\:{r}_{\mathrm{2}} .\:{Here},\:{r}_{\mathrm{1}} >{r}_{\mathrm{2}} \:{and}\:{h}_{\mathrm{1}} >{h}_{\mathrm{2}} . \\ $$$${The}\:{axis}\:{of}\:{each}\:{cylindrical}\:{body}, \\ $$$${passing}\:{through}\:{their}\:{respective} \\ $$$${centre}\:{of}\:{mass}\:{and}\:{perpendicular}\:{to} \\ $$$${their}\:{circular}\:{faces},\:{all}\:{coincide}\:{with} \\ $$$${each}\:{other}.\:{Determine}\:{the}\:\mathrm{3}{D}\:{moment} \\ $$$${of}\:{inertia}\:{of}\:{the}\:{yoyo}\:{about}\:{its}\:{axis} \\ $$$${normal}\:{to}\:{its}\:{outter}\:{circular}\:{faces}\:{and}\:{passing} \\ $$$${through}\:{its}\:{centre}\:{of}\:{mass}. \\ $$$$ \\ $$$$ \\ $$
Commented by prakash jain last updated on 11/Dec/15
                   A                     ↑         ∗    ∗   ∗   ∗    ∗         ∗    ∗   ∗   ∗    ∗                ∗   ∗   ∗      −− −  ∗   ∗   ∗ −−→B                   ∗   ∗   ∗             ∗    ∗   ∗   ∗    ∗         ∗    ∗   ∗   ∗    ∗                      ∣  The yoyo in my diagram is 90 degree rotated.  You intend to calculated moment of intertia  about axis A or B?  Axis A (corrected based on comment below)  m_1  mass of outer cylinder  m_2  mass of inner cylinder  moment of inertia of inner cylider=((m_2 r_2 ^2 )/2)  moment of inertia of outer cylider=((m_1 r_1 ^2 )/2)  Total moment of inertia=m_1 r_1 ^2 +(1/2)m_2 r_1 ^2   Assuming all cylinders of the same density d  M=d(2πr_1 ^2 h_1 +πr_2 ^2 h_2 )⇒d=(M/(π(2r_1 ^2 h_1 +r_2 ^2 h_2 )))  m_1 =((Mr_1 ^2 h_1 )/((2r_1 ^2 h_1 +r_2 ^2 h_2 )))  m_2 =((Mr_2 ^2 h_2 )/((2r_1 ^2 h_1 +r_2 ^2 h_2 )))  I=((Mr_1 ^2 h_1 r_1 ^2 )/((2r_1 ^2 h_1 +r_2 ^2 h_2 )))+((Mr_2 ^2 h_2 r_2 ^2 )/(2(2r_1 ^2 h_1 +r_2 ^2 h_2 )))  =((M(2r_1 ^4 h_1 +r_2 ^4 h_2 ))/(2(2r_1 ^2 h_1 +r_2 ^2 h_2 )))
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{A} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\uparrow \\ $$$$\:\:\:\:\:\:\:\ast\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\ast \\ $$$$\:\:\:\:\:\:\:\ast\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\ast \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\: \\ $$$$−−\:−\:\:\ast\:\:\:\ast\:\:\:\ast\:−−\rightarrow\mathrm{B}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\ast\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\ast \\ $$$$\:\:\:\:\:\:\:\ast\:\:\:\:\ast\:\:\:\ast\:\:\:\ast\:\:\:\:\ast \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mid \\ $$$$\mathrm{The}\:\mathrm{yoyo}\:\mathrm{in}\:\mathrm{my}\:\mathrm{diagram}\:\mathrm{is}\:\mathrm{90}\:\mathrm{degree}\:\mathrm{rotated}. \\ $$$$\mathrm{You}\:\mathrm{intend}\:\mathrm{to}\:\mathrm{calculated}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{intertia} \\ $$$$\mathrm{about}\:\mathrm{axis}\:\mathrm{A}\:\mathrm{or}\:\mathrm{B}? \\ $$$$\mathrm{Axis}\:\mathrm{A}\:\left(\mathrm{corrected}\:\mathrm{based}\:\mathrm{on}\:\mathrm{comment}\:\mathrm{below}\right) \\ $$$${m}_{\mathrm{1}} \:\mathrm{mass}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{cylinder} \\ $$$${m}_{\mathrm{2}} \:\mathrm{mass}\:\mathrm{of}\:\mathrm{inner}\:\mathrm{cylinder} \\ $$$$\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{inner}\:\mathrm{cylider}=\frac{{m}_{\mathrm{2}} {r}_{\mathrm{2}} ^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{cylider}=\frac{{m}_{\mathrm{1}} {r}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\mathrm{Total}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}={m}_{\mathrm{1}} {r}_{\mathrm{1}} ^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{2}} {r}_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$\mathrm{Assuming}\:\mathrm{all}\:\mathrm{cylinders}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{density}\:{d} \\ $$$${M}={d}\left(\mathrm{2}\pi{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +\pi{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)\Rightarrow{d}=\frac{{M}}{\pi\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)} \\ $$$${m}_{\mathrm{1}} =\frac{{Mr}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} }{\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)} \\ $$$${m}_{\mathrm{2}} =\frac{{Mr}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} }{\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)} \\ $$$${I}=\frac{{Mr}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} {r}_{\mathrm{1}} ^{\mathrm{2}} }{\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)}+\frac{{Mr}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} {r}_{\mathrm{2}} ^{\mathrm{2}} }{\mathrm{2}\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)} \\ $$$$=\frac{{M}\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{4}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{4}} {h}_{\mathrm{2}} \right)}{\mathrm{2}\left(\mathrm{2}{r}_{\mathrm{1}} ^{\mathrm{2}} {h}_{\mathrm{1}} +{r}_{\mathrm{2}} ^{\mathrm{2}} {h}_{\mathrm{2}} \right)} \\ $$
Commented by Yozzi last updated on 11/Dec/15
I′d like the M.I about axis A.
$${I}'{d}\:{like}\:{the}\:{M}.{I}\:{about}\:{axis}\:{A}. \\ $$
Commented by Yozzi last updated on 11/Dec/15
Thanks for the calculation. I think  you calculated M.I about axis A.  I′d check if the calculus approach  in 3D yields a different answer.
$${Thanks}\:{for}\:{the}\:{calculation}.\:{I}\:{think} \\ $$$${you}\:{calculated}\:{M}.{I}\:{about}\:{axis}\:{A}. \\ $$$${I}'{d}\:{check}\:{if}\:{the}\:{calculus}\:{approach} \\ $$$${in}\:\mathrm{3}{D}\:{yields}\:{a}\:{different}\:{answer}. \\ $$$$ \\ $$
Commented by prakash jain last updated on 11/Dec/15
Is the assumption about uniform density correct.  I intended to calculate about axis B.
$$\mathrm{Is}\:\mathrm{the}\:\mathrm{assumption}\:\mathrm{about}\:\mathrm{uniform}\:\mathrm{density}\:\mathrm{correct}. \\ $$$$\mathrm{I}\:\mathrm{intended}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{about}\:\mathrm{axis}\:\mathrm{B}. \\ $$
Commented by Yozzi last updated on 11/Dec/15
If we assume uniformity of a regular  body, its mass density is the same at  any observed point of the volume the  body takes. So, yes. This would imply  constant density, given that influencial  environmental factors are   disregarded or invariant in the   given scenario.   Each part could be assigned different  densities also, but for a yoyo it is  required that the outter materials  are identical in composition to  ensure its usual functioning as a toy.  If you wished to calculate M.I for the  body about axis B, which I assume  is parallel to the planes of the outter circular faces,  both the parallel and perpendicular  axis theorems are (possibly) needed.  Obtain the M.I about the axis of one of the  outter cylinders  parallel to its side  length (h_1 ) as I_x .  By the perpendicular axes theorem,  if I_y  is the M.I of the cylinder about an axis parallel  to axis B and I_(z )  is the M.I about  an axis such that axis X,axis Y  and axis Z are mutually perpendicular,  then I_x =I_y +I_z . Since the body is   regular and uniform, I_y =I_z .  ∴I_x =2I_y ⇒I_y =(1/2)I_x .  Let m_1  be the mass of each outter  cylinder. Then, by the parallel axis  theorem, the M.I of each outter   cylinder about axis B is given by  I_b ^((1)) =I_y +(((h_1 +h_2 )/2))^2 m_1 .  I_b ^((1)) =(1/2)I_x +((m_1 (h_1 +h_2 )^2 )/4)  ((h_1 +h_2 )/2) is the ⊥ distance between  axis Y and axis B.  The inner cylinder has I_b ^((2)) =(1/2)I_x ^((1)) .  because of the perpendicular axis  theorem.  The M.I of the full figure about B is  then  I_B =2I_b ^((1)) +I_b ^((2))   I_B =I_x +((m_1 (h_1 +h_2 )^2 )/2)+I_x ^((1)) .  I_x =(1/2)m_1 r_1 ^(2  ) and I_x ^((1)) =(1/2)m_2 r_2 ^2 .
$${If}\:{we}\:{assume}\:{uniformity}\:{of}\:{a}\:{regular} \\ $$$${body},\:{its}\:{mass}\:{density}\:{is}\:{the}\:{same}\:{at} \\ $$$${any}\:{observed}\:{point}\:{of}\:{the}\:{volume}\:{the} \\ $$$${body}\:{takes}.\:{So},\:{yes}.\:{This}\:{would}\:{imply} \\ $$$${constant}\:{density},\:{given}\:{that}\:{influencial} \\ $$$${environmental}\:{factors}\:{are}\: \\ $$$${disregarded}\:{or}\:{invariant}\:{in}\:{the}\: \\ $$$${given}\:{scenario}.\: \\ $$$${Each}\:{part}\:{could}\:{be}\:{assigned}\:{different} \\ $$$${densities}\:{also},\:{but}\:{for}\:{a}\:{yoyo}\:{it}\:{is} \\ $$$${required}\:{that}\:{the}\:{outter}\:{materials} \\ $$$${are}\:{identical}\:{in}\:{composition}\:{to} \\ $$$${ensure}\:{its}\:{usual}\:{functioning}\:{as}\:{a}\:{toy}. \\ $$$${If}\:{you}\:{wished}\:{to}\:{calculate}\:{M}.{I}\:{for}\:{the} \\ $$$${body}\:{about}\:{axis}\:{B},\:{which}\:{I}\:{assume} \\ $$$${is}\:{parallel}\:{to}\:{the}\:{planes}\:{of}\:{the}\:{outter}\:{circular}\:{faces}, \\ $$$${both}\:{the}\:{parallel}\:{and}\:{perpendicular} \\ $$$${axis}\:{theorems}\:{are}\:\left({possibly}\right)\:{needed}. \\ $$$${Obtain}\:{the}\:{M}.{I}\:{about}\:{the}\:{axis}\:{of}\:{one}\:{of}\:{the} \\ $$$${outter}\:{cylinders}\:\:{parallel}\:{to}\:{its}\:{side} \\ $$$${length}\:\left({h}_{\mathrm{1}} \right)\:{as}\:{I}_{{x}} . \\ $$$${By}\:{the}\:{perpendicular}\:{axes}\:{theorem}, \\ $$$${if}\:{I}_{{y}} \:{is}\:{the}\:{M}.{I}\:{of}\:{the}\:{cylinder}\:{about}\:{an}\:{axis}\:{parallel} \\ $$$${to}\:{axis}\:{B}\:{and}\:{I}_{{z}\:} \:{is}\:{the}\:{M}.{I}\:{about} \\ $$$${an}\:{axis}\:{such}\:{that}\:{axis}\:{X},{axis}\:{Y} \\ $$$${and}\:{axis}\:{Z}\:{are}\:{mutually}\:{perpendicular}, \\ $$$${then}\:{I}_{{x}} ={I}_{{y}} +{I}_{{z}} .\:{Since}\:{the}\:{body}\:{is}\: \\ $$$${regular}\:{and}\:{uniform},\:{I}_{{y}} ={I}_{{z}} . \\ $$$$\therefore{I}_{{x}} =\mathrm{2}{I}_{{y}} \Rightarrow{I}_{{y}} =\frac{\mathrm{1}}{\mathrm{2}}{I}_{{x}} . \\ $$$${Let}\:{m}_{\mathrm{1}} \:{be}\:{the}\:{mass}\:{of}\:{each}\:{outter} \\ $$$${cylinder}.\:{Then},\:{by}\:{the}\:{parallel}\:{axis} \\ $$$${theorem},\:{the}\:{M}.{I}\:{of}\:{each}\:{outter}\: \\ $$$${cylinder}\:{about}\:{axis}\:{B}\:{is}\:{given}\:{by} \\ $$$${I}_{{b}} ^{\left(\mathrm{1}\right)} ={I}_{{y}} +\left(\frac{{h}_{\mathrm{1}} +{h}_{\mathrm{2}} }{\mathrm{2}}\right)^{\mathrm{2}} {m}_{\mathrm{1}} . \\ $$$${I}_{{b}} ^{\left(\mathrm{1}\right)} =\frac{\mathrm{1}}{\mathrm{2}}{I}_{{x}} +\frac{{m}_{\mathrm{1}} \left({h}_{\mathrm{1}} +{h}_{\mathrm{2}} \right)^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\frac{{h}_{\mathrm{1}} +{h}_{\mathrm{2}} }{\mathrm{2}}\:{is}\:{the}\:\bot\:{distance}\:{between} \\ $$$${axis}\:{Y}\:{and}\:{axis}\:{B}. \\ $$$${The}\:{inner}\:{cylinder}\:{has}\:{I}_{{b}} ^{\left(\mathrm{2}\right)} =\frac{\mathrm{1}}{\mathrm{2}}{I}_{{x}} ^{\left(\mathrm{1}\right)} . \\ $$$${because}\:{of}\:{the}\:{perpendicular}\:{axis} \\ $$$${theorem}. \\ $$$${The}\:{M}.{I}\:{of}\:{the}\:{full}\:{figure}\:{about}\:{B}\:{is} \\ $$$${then} \\ $$$${I}_{{B}} =\mathrm{2}{I}_{{b}} ^{\left(\mathrm{1}\right)} +{I}_{{b}} ^{\left(\mathrm{2}\right)} \\ $$$${I}_{{B}} ={I}_{{x}} +\frac{{m}_{\mathrm{1}} \left({h}_{\mathrm{1}} +{h}_{\mathrm{2}} \right)^{\mathrm{2}} }{\mathrm{2}}+{I}_{{x}} ^{\left(\mathrm{1}\right)} . \\ $$$${I}_{{x}} =\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{1}} {r}_{\mathrm{1}} ^{\mathrm{2}\:\:} {and}\:{I}_{{x}} ^{\left(\mathrm{1}\right)} =\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{2}} {r}_{\mathrm{2}} ^{\mathrm{2}} . \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by prakash jain last updated on 11/Dec/15
Sorry about the previous comments. The  calculation that I did is applicable for axis A.  Which is normal rotation of yoyo.  You were correct!
$$\mathrm{Sorry}\:\mathrm{about}\:\mathrm{the}\:\mathrm{previous}\:\mathrm{comments}.\:\mathrm{The} \\ $$$$\mathrm{calculation}\:\mathrm{that}\:\mathrm{I}\:\mathrm{did}\:\mathrm{is}\:\mathrm{applicable}\:\mathrm{for}\:\mathrm{axis}\:\mathrm{A}. \\ $$$$\mathrm{Which}\:\mathrm{is}\:\mathrm{normal}\:\mathrm{rotation}\:\mathrm{of}\:\mathrm{yoyo}. \\ $$$$\mathrm{You}\:\mathrm{were}\:\mathrm{correct}! \\ $$
Commented by Yozzi last updated on 11/Dec/15
No problem. I′m hoping to write an  analysis of the mechanics of a yoyo  when subjected to certain scenarios.  Probability also arises since rotational  motion is affected by the occasional  friction between the ordinary string  and the inner ′walls′ of the yoyo as  the wounded string is spun out.   That probability section might be  a bit challenging to formalise.
$${No}\:{problem}.\:{I}'{m}\:{hoping}\:{to}\:{write}\:{an} \\ $$$${analysis}\:{of}\:{the}\:{mechanics}\:{of}\:{a}\:{yoyo} \\ $$$${when}\:{subjected}\:{to}\:{certain}\:{scenarios}. \\ $$$${Probability}\:{also}\:{arises}\:{since}\:{rotational} \\ $$$${motion}\:{is}\:{affected}\:{by}\:{the}\:{occasional} \\ $$$${friction}\:{between}\:{the}\:{ordinary}\:{string} \\ $$$${and}\:{the}\:{inner}\:'{walls}'\:{of}\:{the}\:{yoyo}\:{as} \\ $$$${the}\:{wounded}\:{string}\:{is}\:{spun}\:{out}.\: \\ $$$${That}\:{probability}\:{section}\:{might}\:{be} \\ $$$${a}\:{bit}\:{challenging}\:{to}\:{formalise}. \\ $$
Commented by prakash jain last updated on 12/Dec/15
Nwot related to question directly but when  you are working on a topic are you taking to  a logical conclusion?
$$\mathrm{Nwot}\:\mathrm{related}\:\mathrm{to}\:\mathrm{question}\:\mathrm{directly}\:\mathrm{but}\:\mathrm{when} \\ $$$$\mathrm{you}\:\mathrm{are}\:\mathrm{working}\:\mathrm{on}\:\mathrm{a}\:\mathrm{topic}\:\mathrm{are}\:\mathrm{you}\:\mathrm{taking}\:\mathrm{to} \\ $$$$\mathrm{a}\:\mathrm{logical}\:\mathrm{conclusion}?\: \\ $$
Commented by Yozzi last updated on 12/Dec/15
Well, nowadays I am trying at math  slightly higher than I was taught with  the approach of deductive reasoning  in mind. So yes. I′m searching for  the building blocks of any solution  I see. And many times that   includes knowing and understanding  definitions and my mathematical  knowledge bank is very young. I  hope I could approach number theory  with deductive reasoning when I   formally begin to learn it.
$${Well},\:{nowadays}\:{I}\:{am}\:{trying}\:{at}\:{math} \\ $$$${slightly}\:{higher}\:{than}\:{I}\:{was}\:{taught}\:{with} \\ $$$${the}\:{approach}\:{of}\:{deductive}\:{reasoning} \\ $$$${in}\:{mind}.\:{So}\:{yes}.\:{I}'{m}\:{searching}\:{for} \\ $$$${the}\:{building}\:{blocks}\:{of}\:{any}\:{solution} \\ $$$${I}\:{see}.\:{And}\:{many}\:{times}\:{that}\: \\ $$$${includes}\:{knowing}\:{and}\:{understanding} \\ $$$${definitions}\:{and}\:{my}\:{mathematical} \\ $$$${knowledge}\:{bank}\:{is}\:{very}\:{young}.\:{I} \\ $$$${hope}\:{I}\:{could}\:{approach}\:{number}\:{theory} \\ $$$${with}\:{deductive}\:{reasoning}\:{when}\:{I}\: \\ $$$${formally}\:{begin}\:{to}\:{learn}\:{it}.\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Answered by prakash jain last updated on 11/Dec/15
For axis B  inertia of inner cylinder=(m_2 /(12))(3r_2 ^2 +h_2 ^2 )  inertia of outer cylinder =(m_1 /(12))(3r_1 ^2 +h_1 ^2 )      for axis ∥ to faces of cylinder.  Also distance of axis B from center  of gravity=((h_1 +h_2 )/2) (this is same for both cylinders)  inertia of outer cylinders (axis B)=       m_1 (((3r_1 ^2 +h_1 ^2 )/(12))+(((h_1 +h_2 )^2 )/4))  Total moment of inertia      =2m_1 (((3r_1 ^2 +h_1 ^2 )/(12))+(((h_1 +h_2 )^2 )/4))+(m_2 /(12))(3r_2 ^2 +h_2 ^2 )
$$\mathrm{For}\:\mathrm{axis}\:\mathrm{B} \\ $$$$\mathrm{inertia}\:\mathrm{of}\:\mathrm{inner}\:\mathrm{cylinder}=\frac{{m}_{\mathrm{2}} }{\mathrm{12}}\left(\mathrm{3}{r}_{\mathrm{2}} ^{\mathrm{2}} +{h}_{\mathrm{2}} ^{\mathrm{2}} \right) \\ $$$$\mathrm{inertia}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{cylinder}\:=\frac{{m}_{\mathrm{1}} }{\mathrm{12}}\left(\mathrm{3}{r}_{\mathrm{1}} ^{\mathrm{2}} +{h}_{\mathrm{1}} ^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\mathrm{for}\:\mathrm{axis}\:\parallel\:\mathrm{to}\:\mathrm{faces}\:\mathrm{of}\:\mathrm{cylinder}. \\ $$$$\mathrm{Also}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{axis}\:\mathrm{B}\:\mathrm{from}\:\mathrm{center} \\ $$$$\mathrm{of}\:\mathrm{gravity}=\frac{{h}_{\mathrm{1}} +{h}_{\mathrm{2}} }{\mathrm{2}}\:\left(\mathrm{this}\:\mathrm{is}\:\mathrm{same}\:\mathrm{for}\:\mathrm{both}\:\mathrm{cylinders}\right) \\ $$$$\mathrm{inertia}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{cylinders}\:\left(\mathrm{axis}\:\mathrm{B}\right)= \\ $$$$\:\:\:\:\:{m}_{\mathrm{1}} \left(\frac{\mathrm{3}{r}_{\mathrm{1}} ^{\mathrm{2}} +{h}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{12}}+\frac{\left({h}_{\mathrm{1}} +{h}_{\mathrm{2}} \right)^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$\mathrm{Total}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia} \\ $$$$\:\:\:\:=\mathrm{2}{m}_{\mathrm{1}} \left(\frac{\mathrm{3}{r}_{\mathrm{1}} ^{\mathrm{2}} +{h}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{12}}+\frac{\left({h}_{\mathrm{1}} +{h}_{\mathrm{2}} \right)^{\mathrm{2}} }{\mathrm{4}}\right)+\frac{{m}_{\mathrm{2}} }{\mathrm{12}}\left(\mathrm{3}{r}_{\mathrm{2}} ^{\mathrm{2}} +{h}_{\mathrm{2}} ^{\mathrm{2}} \right) \\ $$

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