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Find the moment of inertia of the system shown in the figure about the axis shown ?( Ans : 461 kg metre square) <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 415 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-520092929 1073786111 9 0 415 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt; mso-ascii-font-family:Calibri; mso-fareast-font-family:Calibri; mso-hansi-font-family:Calibri;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:875776197; mso-list-type:hybrid; mso-list-template-ids:-73264390 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} @list l0:level1 {mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in;} @list l1 {mso-list-id:914319462; mso-list-type:hybrid; mso-list-template-ids:-24851924 557222636 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} @list l1:level1 {mso-level-number-format:alpha-lower; mso-level-text:"%1\)"; mso-level-tab-stop:none; mso-level-number-position:left; margin-left:.75in; text-indent:-.25in;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> 2. A rod of mass m and length l is released from height h as shown in the figure .The collision between rod and point A is elastic whereas the coefficient of restitution for collision between rod and B is ½ , then find a) Velocity of centre of mass of rod just after collision .( Ans: ( (9gh)/8) ^ ½ ) b) Angular velocity of rod just after collision.( Ans: (gh)/(2l*l) ^ ½ ) c) Maximum height gain by the centre of the rod after collision.( Ans: 9h/16) d) Angular velocity of the rod at the maximum height .( Ans: (2gh) ^ ½ / 2l ) <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 415 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-520092929 1073786111 9 0 415 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt; mso-ascii-font-family:Calibri; mso-fareast-font-family:Calibri; mso-hansi-font-family:Calibri;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:875776197; mso-list-type:hybrid; mso-list-template-ids:-73264390 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} @list l0:level1 {mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> 3. Four thin rods each of mass m and length l is joined to form a square .Find the moment of inertia of the loop about the axis shown in the figure? (Ans: 2m*l*l/3 ) <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 415 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-520092929 1073786111 9 0 415 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:.5in; mso-add-space:auto; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman";} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt; mso-ascii-font-family:Calibri; mso-fareast-font-family:Calibri; mso-hansi-font-family:Calibri;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:875776197; mso-list-type:hybrid; mso-list-template-ids:-73264390 67698703 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} @list l0:level1 {mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in;} @list l1 {mso-list-id:1002397323; mso-list-type:hybrid; mso-list-template-ids:622353104 400199396 67698713 67698715 67698703 67698713 67698715 67698703 67698713 67698715;} @list l1:level1 {mso-level-number-format:alpha-lower; mso-level-text:"%1\)"; mso-level-tab-stop:none; mso-level-number-position:left; margin-left:.75in; text-indent:-.25in;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> 4. A small disc and a thin uniform rod length L , whose mass is n times greater than the mass of the disc , lie on a smooth horizontal plane.The disc is set in motion , in horizontal direction and perpendicular to the rod , with velocity v , after which it elastically collides with the end of the rod. a) Find the velocity of the disc ? (Ans : v(4-n)/(4+n) ) b) Find the angular velocity of the rod after the collision ? ( Ans : 12v/( (4+n)*L) ) c) At what value of n will the velocity of the disc after the collision reverses the direction ?( ans: n>4 )

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1.       Find the moment of inertia  of the system shown in the figure  about the axis shown ?( Ans : 461 kg metre square)

                                

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2.       A  rod of mass m and length  l is released  from height  h as shown in the figure .The collision between rod and point A  is elastic  whereas  the coefficient  of restitution  for  collision  between  rod  and B is ½ , then find

a)      Velocity  of centre of mass  of rod  just after collision .( Ans:   ( (9gh)/8)^ ½    )

b)      Angular velocity  of rod  just after collision.( Ans:   (gh)/(2l*l)^ ½  )

c)       Maximum height gain by the centre of the rod after collision.( Ans:   9h/16)

d)      Angular velocity  of the rod  at the maximum  height .(   Ans:  (2gh) ^ ½  / 2l    )

 

                                    

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3.       Four thin rods  each of mass m and length l is joined to form a square .Find the moment of inertia  of the loop  about the axis shown in the figure?  (Ans: 2m*l*l/3 )

 

                 

 

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4.       A small  disc  and a thin  uniform rod length L , whose mass is n times greater  than  the mass of the disc , lie  on a smooth horizontal plane.The disc is set in motion , in horizontal direction and perpendicular  to the  rod , with  velocity v , after which it elastically  collides  with the  end of the rod.

a)      Find the velocity of the disc ? (Ans :  v(4-n)/(4+n)  )

b)      Find the angular velocity of the rod after the collision ? ( Ans : 12v/( (4+n)*L)  )

c)       At what value of n will the velocity of the disc  after the collision  reverses the direction ?( ans:  n>4 )