RIGID BODIES - MOMENT OF INERTIA

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Shonda King
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1 IID DIES - ET F IETI The inabiity of a body to change by itsef its position of rest or uniform motion is caed Inertia. The greater the mass of the body, the greater its inertia as greater force is required to bring about a desired change in the body. Thus the mass of the body is taken as a measure of its inertia for transatory motion. Simiary a body, capabe of rotation about an ais, possesses inertia for rotationa motion. The greater the coupe or torque required to change the state of rotation of the body, the greater its rotationa inertia. This rotationa inertia of the body is caed the.i of the body about the ais of rotation. Thus.I. is the rotationa amaogue of mass in transatory motion. Definition of.i of a partice of mass m about a given ais of rotation is defined as the product of the mass and the square of the distance r of the partice from the ais of rotation. i.e.,.i I r Consider a body of mass capabe of rotation about a fied ais. The body can be imagined to be made up of a no. of partices of mass m 1, m, m,... at distances r 1, r, r,... from the ais of rotation. Then the.i of the body about the ais of rotation sum of the.is of a the partices about the ais of rotation i.e., I m r + m r + m r +... Σmr 1 1 I mr Σ -----(1) r 1 r K m 1 m C r m Thus.I of a body about an ais of rotation is defined as the sum of the products of the mass and square of the distance of a the partices constituting the body from the ais of rotation. DIES F TI Consider a body of mass capabe of rotation about an ais. et the entire mass of the body be imagined to be concentrated at a point C. This point is caed the Centre of mass of the body. et C be at a distance K from the ais of rotation. Then.I. of the body about the ais of rotation can be written as I K ---() where K is caed the radius of gyration of the body w.r.t. the ais of rotation. Def : adius of gyration of a body capabe of rotation about an ais is defined as the distance of the point where the entire mass of the body is imagined to be concentrated from the ais of rotation. From (1) & () : K Σmr or K mr KIETIC EE F TTI Consider a rigid body rotating with a constant anguar speed ω about a fied ais as shwon in the figure. s the body is made up of a number of partices of masses m 1, m, m,... at distances r 1, r, r,... from the ais of rotation, a these partices describe circuar paths of radii r 1, r, r,... with the same anguar speed ω. s the inear veocity of the partices V rω, V is different for different partices. et V 1, V, V,... be the inear veocities of the partices. Then the tota K.E. of rotation of the body, E 1 m v 1 m v m v

2 1 m r ω + m r ω ω m r + m r +m r ω Ι, where I is the I of the body about the ais rotation. \ E 1 Iω THEE F PE ES Statement : The.I of a body about any ais is equa to the sum of the.i of the body about a parae ais passing through its C.. and the product of the mass of the body and square of the distance b/w the two aes. This is caed Steiner s theorem. Proof : Consider a body of mass whose C.. is at. et the body rotate about an ais and its.i about about be I. et CD be a parae ais passing through. et the separation b/w the two aes be r. Consider a partice of mass m at P at a distance from CD. Then.I of this partice about m(r + ).I of the whoe body about is I Σ m(r + ) Σ mr + Σ m + Σ mr r + I + r Σ m...(1), where Σ m is the tota mass of the body, Σ m I is the.i of the body about CD. The weight of the partice of mass m at P force acting on m mg. The moment of this force about CD mg. Sum of the moments of the weights of a the partices about CD Σmg. The agebraic sum of the moments of a the forces about an ais passing through the centre of gravity of a body. \ Σmg g, Σm \ r Σm -----() Substituting () in (1), I I + r Hence the theorem r C D P THEE F PEPEDICU ES Z Statement : The.I of a amina about any ais r to its surface is equa to the sum of the moments of inertia about two r aes in the pane of the amina, a the three aes passing through the same point on the amina. et & be two rectanguar aes in the pane of the amina and Z, an ais through ` r to both &.. Consider a partice of mass m at a point ρ distant `r from in the - pane. et its coordinates be & y. Then r + y. r m p(, z)

3 .I. of the partice about z-ais mr..i of the amina about z-ais, I Z Σ mr. I Z Σ m( + y ) Σ m.i of the amina about y-ais I y Σ my.i of the amina about -ais I. Hence the theorem. THEE F PEPEDICU ES I THEE DIESIS et, & Z be three rectanguar aes with the origin at in the given body. Consider a partice of mass m at a point P. et P r. From P draw P ^r to the pane. Draw parae to & parae to. Then is the - coordinate and y is the y-coordinate of P. Join, is ^r to Z. Draw P parae to. Then P is aso ^r to Z. P z is the Z- coordinate of P. and r + y + z (1).I of the partice at P about the Z-ais m P m( + ) m P m m( + y ).I of the body about the Z-ais, I Z Σ m( + y ) I Z Σ m + Σ my -----() y I Σ m + Σ mz -----() and I Z Σ my + Σ mz -----() I + I + I Z Σ my + Σ mz + Σ m + Σ mz + Σ m + Σ my Σ m + Σ my + Σ mz Σ m( + y + z ) I + I y + I Z Σ mr {from (1) I + I y + I Z I d 1 or Ι Ι + Ι + Ι y z i Thus the.i of a three dimensiona body about any ais passing through a point in the body is equa to haf the sum of the.i of the body about three mutuay ^r aes passing through the same point. y Z r z p(, y, z) y 1a).I of a thin rod about an ais perpendicuar to its ength and passing through its C.. Consider a thin uniform rod of ength and mass, then its mass per unit ength m. et be an ais ^r to its ength and passing through its C..`. Consider a sma eement of thickness d at a distance from the ais of rotation. Then mass of the eement md..i of this eement about md m d \.I of the rod about is

4 z Ι m d m P P m + m m If K is the rodius of gyration of the rod about the ais of rotation, then I K \ K K 1 1 b).i of the rod about an ais at one end of the rod and perpendicuar to the rod et be an ais at one end and ^r to the ength of the rod. z m m m Then I m d P \ I. We know I K. \ K K d..i of a rectanguar amina (a) bout an ais passing through its centre and parae to breadth Consider a rectanguar amina CD of mass, ength and breadth b. et m be its mass per unit area i.e., m, or m( b). b / is an ais e to its breadth. Consider a sma strip of thickness d at a distance from the ais. Its mass m (b d).i of the strip about / mbd mb d \.I of the rectanguar amina about / is Ι mb zmb d F H mb I KJ mb b 8 ' D m( b) 1 1 ' d C I 1

5 b) is Passing through its Centre and parae to ength : / is an ais passing through its centre and e to the ength. Consider a sma strip of thickness d at a distance from the ais. Its mass m ( d)..i of the strip about / m d m d \.I of the rectanguar amina about / is b z Ι m b d m m b m( b) b I 8 1 d 1 b Ι m b b m b 8 m( b) b 1 b 1 mb I 1 c) is Perpendicuar to the pane of the amina and passing through ccording to the theorem of ^r aes,.i of the amina about an ais ^r to the pane of the amina and passing through.i about two mutuay ^r aes through in the pane of the amina. et I be the.i of the amina about an ais ZZ / passing through its C and ^r to its pane. i.e., Ι Ι + Ι b ( +b ) Ι 1 (d) is aong one end (D) z mb mb Ι m b d I b g Z.I about : b z I m d / D / Z / C 5

6 mb b mb b I b e) is passing through the mid point of D or C and perpendicuar to the pane of the amina CD is the amina. et be the mid point of D; et be an ais through ^r to the pane of the amina. et.i of the amina about be I, / is a e ais through the centre of the amina and to its pane. et the.i of the amina about / be I. ( + b ) Then Ι 1 y e aes theorem, I Ι F + H I K J I ( + b ) b I + b 1 F I H K J y.i about an ais through the mid point of or CD and ^r D to the pane of the amina / C b Ι + 1..I of a circuar ring about an ais a) through its centre and perpendicuar to its pane Consider a thin circuar ring of mass and radius. et / be the ais through its centre and ^r to its pane. Consider a sma eement of mass m. The.I of this eement about / m..i of the ring about /, I Σm. I m Σ m and is the distance of each of the eements from / b).i of the ring about an ais aong its diameter et / and / be two aes aong two diameters of the ring. Then I I.I of the ring is the same about a diameters. et I and I be the.i of the ring about / and / respectivey. Then I I. et the.i of the ring about an ais through and ^r to its pane be I. Then I Μ where is 6

7 the mass and is the radius of the ring. y the theorem of ^r aes, I I + I I I I. I I Iy. / \.I of the ring about its diameter / ).I of a circur amina (disc) about an ais passing through its centre and perpendicuar to its pane. et / be an ais through its centre and ^r to its pane. Consider a circuar amina of mass and radius. rea of the amina π. ass per unit area of the amina, m π The disc can be considered to be made up of a number of annuar rings. Consider one such ring of thickness d and radius. rea of the ring π d ass of the ring π d m πm d..i of the ring about / π m d.i of the amina about / Ι π m d z πm d is. Fig.(a) / Ι πm P πm d I (π m) I Fig.(b) b).i about its diameter et / and / be two ^r aes aong two ^r diameters of the disc. et I and I be the.is of the disc about / and / respectivey Then I I..I of the disc about an ais passing through its centre and ^r to its pane, I y the theorem of ^r aes, I I + I I I I I. 7

8 \ I I c).i. about a tangent et be an ais aong a tangent to the disc. et the.i of the disc about be I. et / be an ais parae to and aong the diameter (through the C. ). et the.i of the disc about / be I. Then I is the radius of the disc. y the theorem of parae aes, I I + /, where is the mass and / + 5 I 5).I of a thin sphirica she about diameter Consider a thin spherica she of a mass and radius. et / be an ais aong a diameter of the she. rea of the she π. \ ass per unit area, m π The she may be considered to be made up of a number of rings. et PS be one such ring of thickness d and radius y at a distance from /. rea of the ring πy d From the figure, sinθ y y sinθ. / cosθ cosθ ; d sinθ dθ. P d dθ and + y y. rea of the ring π ( sinθ)( dθ) π ( sinθ dθ) π d (in magnitude) ass of the ring π d m π m d.i of the ring about / (diameter) ass (radius) π m d y π m d ( )..I of the she about / is z I π m ( ) d / dθ θ / P S 8

9 z π m d π m d π m z π m m π π m π m I π m b).i of a thin spherica she about a tangent et.i of the she about a tangent be I and about a e ais aong its diameter (i.e., through its centre of gravity be I.) Then by the theorem of parae aes, I I +, where is the mass of she and is its radius. + 5 I a).i. of a soid sphere about its diameter Consider a soid sphere of radius and mass. et ρ be the density of the materia of the sphere. Voume of the sphere π ass of the sphere π ρ The soid sphere may be considered to be made up of a number of concentric spherica shes of different radii. Consider one such she of radius and thickness d. Surface area of the she π. Voume of the she ass of the she, π d vo. density π d ρ.i of the she about / ass (radius) π d ρ 9

10 8 π ρ d.i of the sphere about / 8, I z π ρ d e j 8 π ρ F 5 π ρ H I K J 5 5 d I 5 / b).i about a tangent y parae aes theorem,.i about the tangent, I I I 5.I. of a soid cyinder about an ais passing through its centre and ^r to its ength Consider a soid cyinder of mass, radius and ength. et / be an ais passing through its centre of gravity and ^r to its ength. et m be the mass per unit ength of the cyinder i.e., m. / The cyinder may be considered to be made up of a number of thin discs of rod. / Consider one such disc of thickness d at a distance from /. ass of the disc md d / /.I of the disc about an ais / aong its own diameter ass (r d) d d.i of the disc about /.I of the disc about / + ass (by e aes theorem) 1

11 d + d d +.I of the cyinder about / is given by d d + d P I z d + d P P I + 1 P b).i about the diameter of one of its faces I.I about / I + ass H K J P I P F I + P / 11

12 c).i about its own ais.i of the disc about an ais pasisng through its centre and ^r to its pane.i of the disc about the ais / of the cyinder ass of the disc rad.i of the cyinder about / is I b g m Ι Σm I 1

Previous Years Problems on System of Particles and Rotional Motion for NEET

Previous Years Problems on System of Particles and Rotional Motion for NEET P-8 JPME Topicwise Soved Paper- PHYSCS Previous Years Probems on Sstem of Partices and otiona Motion for NEET This Chapter Previous Years Probems on Sstem of Partices and otiona Motion for NEET is taken