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physics

mechanics

Easy

Question

A metal disc of radius R and mass M freely rolls down from the top of an inclined plane of height h without slipping. The speed of its centre of mass on reaching the bottom of the inclined plane is

$\sqrt{\frac{4 g h}{3}}$
$\sqrt{\frac{3 g h}{4}}$
$\sqrt{g h}$
$\sqrt{\frac{g h}{2}}$

The correct answer is: $\sqrt{\frac{4 g h}{3}}$

$text For disc end text not stretchy rightwards arrow beta equals 1 half table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell text By L.C.E. end text end cell row cell M g h equals 1 half M V subscript C M end subscript superscript 2 left parenthesis 1 plus beta right parenthesis ∣ end cell row cell g h equals 1 half V subscript c m end subscript superscript 2 open parentheses 1 plus 1 half close parentheses end cell row cell g h equals 1 half V subscript c m end subscript superscript 2 open parentheses 3 over 2 close parentheses end cell row cell V subscript c m end subscript superscript 2 equals fraction numerator 4 g h over denominator 3 end fraction not stretchy rightwards double arrow V subscript c m end subscript equals square root of fraction numerator 4 g h over denominator 3 end fraction end root end cell end table$

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

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

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

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

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

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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

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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

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

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

Mechanics

Physics

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

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

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