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Physics

Mechanics

Easy

Question

A thin wire of length L and uniform linear mass density is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX’ is :

$\frac{ρ L^{3}}{8 π^{2}}$
$\frac{ρ L^{3}}{16 π^{2}}$
$\frac{5 ρ L^{3}}{16 π^{2}}$
$\frac{3 ρ L^{3}}{8 π^{2}}$

The correct answer is: $fraction numerator 3 rho L cubed over denominator 8 pi squared end fraction$

$I equals m R squared plus 1 half m R squared equals 3 over 2 m R squared equals 3 over 2 rho l open parentheses fraction numerator l over denominator 2 pi end fraction close parentheses squared equals fraction numerator 3 rho l cubed over denominator 8 pi squared end fraction$

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$straight A space uniform space rod space of space mass space straight M equals 2 kg space and space length space straight L space is space suspended space by space two space smooth space hinges space 1 space and space 2 space as space shown space in space figure. space straight A space force space straight F equals 4 straight N space is space applied space downward space at space straight a space distance space straight L divided by 4 space from space hinge space 2. space Due space to space the space application space of space force space straight F comma space hinge space 2 space breaks. space At space this space instant space comma space applied space force space straight F space is space also space removed. space The space rod space starts space to space rotate space downward space about space hinge space 1. space open parentheses g equals 10 m divided by s squared close parentheses$

$The space reaction space at space hinge space 1 comma space just space after space breaking space of space hinge space 2 comma space is space$

$straight A space uniform space rod space of space mass space straight M equals 2 kg space and space length space straight L space is space suspended space by space two space smooth space hinges space 1 space and space 2 space as space shown space in space figure. space straight A space force space straight F equals 4 straight N space is space applied space downward space at space straight a space distance space straight L divided by 4 space from space hinge space 2. space Due space to space the space application space of space force space straight F comma space hinge space 2 space breaks. space At space this space instant space comma space applied space force space straight F space is space also space removed. space The space rod space starts space to space rotate space downward space about space hinge space 1. space open parentheses g equals 10 m divided by s squared close parentheses$

$The space reaction space at space hinge space 1 comma space just space after space breaking space of space hinge space 2 comma space is space$

PhysicsMechanics

parallel

$straight A space uniform space rod space of space mass space straight M equals 2 kg space and space length space straight L space is space suspended space by space two space smooth space hinges space 1 space and space 2 space as space shown space in space figure. space straight A space force space straight F equals 4 straight N space is space applied space downward space at space straight a space distance space straight L divided by 4 space from space hinge space 2. space Due space to space the space application space of space force space straight F comma space hinge space 2 space breaks. space At space this space instant space comma space applied space force space straight F space is space also space removed. space The space rod space starts space to space rotate space downward space about space hinge space 1. space open parentheses g equals 10 m divided by s squared close parentheses$

$The space reaction space at space hinge space 1 comma space before space space hinge space 2 space breaks comma space is space$

$straight A space uniform space rod space of space mass space straight M equals 2 kg space and space length space straight L space is space suspended space by space two space smooth space hinges space 1 space and space 2 space as space shown space in space figure. space straight A space force space straight F equals 4 straight N space is space applied space downward space at space straight a space distance space straight L divided by 4 space from space hinge space 2. space Due space to space the space application space of space force space straight F comma space hinge space 2 space breaks. space At space this space instant space comma space applied space force space straight F space is space also space removed. space The space rod space starts space to space rotate space downward space about space hinge space 1. space open parentheses g equals 10 m divided by s squared close parentheses$

$The space reaction space at space hinge space 1 comma space before space space hinge space 2 space breaks comma space is space$

PhysicsMechanics

$straight A space body space of space mass space straight M space and space radius space straight R space is space as space shown space in space figure space. space The space body space can space be space straight a space ring space comma space straight a space disc comma space straight a space hollow space sphere space or space straight a space solid space I subscript x comma I subscript y. space space and space I subscript z space space are space the space moments space of space inertia space about space straight x comma straight y space and space straight z space axis space respectively. space Origin space coincides space with space centre space of space mass space of space the space body.$

Column I
Column II

i. $I subscript x equals I subscript y equals I subscript z$
a. possible for ring or disc

ii. $I subscript x equals I subscript y plus I subscript z$
b. possible for hollow sphere or solid sphere

iii. $I subscript y equals I subscript z plus I subscript x$
c. possible for all of the bodies mentioned

iv. $I subscript z equals I subscript x plus I subscript y$
d. not possible for any of the body mentioned

$straight A space body space of space mass space straight M space and space radius space straight R space is space as space shown space in space figure space. space The space body space can space be space straight a space ring space comma space straight a space disc comma space straight a space hollow space sphere space or space straight a space solid space I subscript x comma I subscript y. space space and space I subscript z space space are space the space moments space of space inertia space about space straight x comma straight y space and space straight z space axis space respectively. space Origin space coincides space with space centre space of space mass space of space the space body.$

Column I
Column II

i. $I subscript x equals I subscript y equals I subscript z$
a. possible for ring or disc

ii. $I subscript x equals I subscript y plus I subscript z$
b. possible for hollow sphere or solid sphere

iii. $I subscript y equals I subscript z plus I subscript x$
c. possible for all of the bodies mentioned

iv. $I subscript z equals I subscript x plus I subscript y$
d. not possible for any of the body mentioned

PhysicsMechanics

Two particles A and B of mass m each, are joined by a rigid massless rod of length l. A particle P of mass m, moving with speed u normal to AB, strikes A and stick to it. The centre of mass of the 'A + B + P' system after collision is C. If u = 4m/s, m = 3 kg, l = 3 m, then (All quantities in column are in SI units)

Two particles A and B of mass m each, are joined by a rigid massless rod of length l. A particle P of mass m, moving with speed u normal to AB, strikes A and stick to it. The centre of mass of the 'A + B + P' system after collision is C. If u = 4m/s, m = 3 kg, l = 3 m, then (All quantities in column are in SI units)

PhysicsMechanics

parallel

$In space Column space straight I comma space information space about space the space force left parenthesis straight s right parenthesis space acting space on space straight a space body space is space mentioned comma space while space in space Column space II space information space about space the space motion space of space straight a space body space is space given. space Match space the space entries space in space Column space II space with space the space entries space in space Column space straight I.$

Column I
Column II

i. A single force through centre of mass
a. Rotational motion

ii.Equal and opposite forces separated by non-zero distance
b. Translation motion

iii. Equal and opposite forces acting at the same point
c. No motion

iv. A single force not through centre of mass
d. Centre of mass performs translatory motion

$In space Column space straight I comma space information space about space the space force left parenthesis straight s right parenthesis space acting space on space straight a space body space is space mentioned comma space while space in space Column space II space information space about space the space motion space of space straight a space body space is space given. space Match space the space entries space in space Column space II space with space the space entries space in space Column space straight I.$

Column I
Column II

i. A single force through centre of mass
a. Rotational motion

ii.Equal and opposite forces separated by non-zero distance
b. Translation motion

iii. Equal and opposite forces acting at the same point
c. No motion

iv. A single force not through centre of mass
d. Centre of mass performs translatory motion

PhysicsMechanics

$From space straight a space uniform space disc space of space mass space straight M space and space radius space straight R comma space straight a space concentric space disc space of space radius space straight R divided by 2 space is space cut space out. space For space the space remaining space space annular space disc colon I subscript 1 space space space is space the space moment space of space inertia space about space axis space ‘ 1 ’ comma space space I subscript 2 space about space ‘ 2 ’ comma space space space space I subscript 3 italic space about space space ‘ 3 ’ comma space and space I subscript 4 space space space about space ‘ 4 ’. space Axes space ‘ 1 ’ space and space ‘ 2 ’ space are space perpendicular space to space the space disc space and space ‘ 3 ’ space and space ‘ 4 ’ space are space in space the space plane space of space the space disc. space Axes space ‘ 2 ’ comma ’ 3 ’ space and space ‘ 4 ’ space intersect space at space straight a space common space point. space Match space the space following space$

Column – I

Column – II

i. $I subscript 1$ is equal to
a. $21 over 32 M R squared$

ii. $I subscript 2$ is equal to
b. $I subscript 1 divided by 2$

iii. $I subscript 3 plus I subscript 4$ is equal to
c. $15 over 32 M R h 2$

iv. $I subscript 2 minus I subscript 3$ is equal to
d. none of these

$From space straight a space uniform space disc space of space mass space straight M space and space radius space straight R comma space straight a space concentric space disc space of space radius space straight R divided by 2 space is space cut space out. space For space the space remaining space space annular space disc colon I subscript 1 space space space is space the space moment space of space inertia space about space axis space ‘ 1 ’ comma space space I subscript 2 space about space ‘ 2 ’ comma space space space space I subscript 3 italic space about space space ‘ 3 ’ comma space and space I subscript 4 space space space about space ‘ 4 ’. space Axes space ‘ 1 ’ space and space ‘ 2 ’ space are space perpendicular space to space the space disc space and space ‘ 3 ’ space and space ‘ 4 ’ space are space in space the space plane space of space the space disc. space Axes space ‘ 2 ’ comma ’ 3 ’ space and space ‘ 4 ’ space intersect space at space straight a space common space point. space Match space the space following space$

Column – I

Column – II

i. $I subscript 1$ is equal to
a. $21 over 32 M R squared$

ii. $I subscript 2$ is equal to
b. $I subscript 1 divided by 2$

iii. $I subscript 3 plus I subscript 4$ is equal to
c. $15 over 32 M R h 2$

iv. $I subscript 2 minus I subscript 3$ is equal to
d. none of these

PhysicsMechanics

$End space straight A space space of space straight a space rod space AB space is space being space pulled space on space the space floor space with space straight a space constant space velocity space space v subscript 0 space as space shown. space Taking space the space length space of the space rod space as space l space comma space at space an space instant space when space the space rod space makes space an space angle space 37 to the power of 0 space space with space the space horizontal comma space calculate space space$

$The space angular space velocity space of space the space end space$

$End space straight A space space of space straight a space rod space AB space is space being space pulled space on space the space floor space with space straight a space constant space velocity space space v subscript 0 space as space shown. space Taking space the space length space of the space rod space as space l space comma space at space an space instant space when space the space rod space makes space an space angle space 37 to the power of 0 space space with space the space horizontal comma space calculate space space$

$The space angular space velocity space of space the space end space$

PhysicsMechanics

parallel

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