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Motion of System of Particles and Rigid Body

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Q 1. From a solid sphere of mass {tex} \mathrm { M } {/tex} and radius {tex} \mathrm { R } , {/tex} a cube of maximum possible volume is cut. Moment of inertia of cube
about an axis passing through its center and perpendicular to one of its faces is :

{tex} \frac { 4 \mathrm { MR } ^ { 2 } } { 9 \sqrt { 3 } \pi } {/tex}

B

{tex} \frac { 4 \mathrm { MR } ^ { 2 } } { 3 \sqrt { 3 } \pi } {/tex}

C

{tex} \frac { \mathrm { MR } ^ { 2 } } { 32 \sqrt { 2 } \pi } {/tex}

D

{tex} \frac { \mathrm { MR } ^ { 2 } } { 16 \sqrt { 2 } \pi } {/tex}

Explanation

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Q 2. A hollow sphere is held suspended. Sand is now poured into it in stages. The centre of gravity of the sphere with the sand

A

rises continuously

B

remains unchanged in the process

C

first rises and then falls to the original position

first falls and then rises to the original position

Explanation

Initially centre of gravity is at the centre. When sand is poured it will fall and again after a limit, centre of gravity will rise.

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Q 3. A body {tex} A {/tex} of mass {tex} M {/tex} while falling vertically downwards under gravity breaks into two parts; a body {tex} B {/tex} of mass {tex} \frac { 1 } { 3 } M {/tex}
and a body {tex} C {/tex} of mass {tex} \frac { 2 } { 3 } M {/tex} . The centre of mass of bodies {tex} B {/tex} and {tex} C {/tex} taken together shifts compared to that of body {tex} A {/tex} towards

does not shift

B

depends on height of breaking

C

body {tex} B {/tex}

D

body {tex} C {/tex}

Explanation

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Q 4. From a uniform wire, two circular loops are made (i) {tex} \mathrm { P } {/tex} of radius {tex} \mathrm { r } {/tex} and (ii) {tex} \mathrm { Q } {/tex} of radius {tex} \mathrm { nr } {/tex}. If the moment of inertia of {tex} \mathrm { Q } {/tex} about an axis passing through its centre and perpendicular to its plane is 8 times that of {tex} \mathrm { P } {/tex} about a similar axis, the value of {tex} \mathrm { n } {/tex} is (diameter of the wire is very much smaller than {tex} \mathrm { r } {/tex} or {tex} \mathrm { nr } {/tex} )

A

8

B

6

C

4

2

Explanation



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Q 5. A billiard ball of mass {tex} m {/tex} and radius {tex} r , {/tex} when hit in a horizontal direction by a cue at a height {tex} h {/tex} above its centre, acquired a
linear velocity {tex} v _ { 0 } . {/tex} The angular velocity {tex} \omega _ { 0 } {/tex} acquired by the ball is

A

{tex} \frac { 5 v _ { 0 } r ^ { 2 } } { 2 h } {/tex}

B

{tex} \frac { 2 v _ { 0 } r ^ { 2 } } { 5 h } {/tex}

C

{tex} \frac { 2 v _ { 0 } h } { 5 r ^ { 2 } } {/tex}

{tex} \frac { 5 v _ { 0 } h } { 2 r ^ { 2 } } {/tex}

Explanation



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Q 6. Two discs of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio of {tex} 1 : 3 . {/tex} The moments of inertia of these discs about the respective axes passing through their centres and perpendicular to their planes will be in the ratio of

A

{tex} 1 : 3 {/tex}

{tex} 3 : 1 {/tex}

C

{tex} 1 : 9 {/tex}

D

{tex} 9 : 1 {/tex}

Explanation





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Q 7. A pulley fixed to the ceiling carries a string with blocks of mass {tex} \mathrm { m } {/tex} and 3{tex} \mathrm { m } {/tex} attached to its ends. The masses of string and pulley are negligible. When the system is released, its centre of mass moves with what acceleration?

A

0

{tex} - g / 4 {/tex}

C

{tex} g / 2 {/tex}

D

{tex} - g / 2 {/tex}

Explanation

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Q 8. A ring of mass {tex} m {/tex} and radius {tex} R {/tex} has four particles each of mass {tex} m {/tex} attached to the ring as shown in figure. The centre of ring has a speed {tex} v _ { 0 } {/tex} . The kinetic energy of the system is

A

{tex} m v _ { 0 } ^ { 2 } {/tex}

B

3{tex} m v _ { 0 } ^ { 2 } {/tex}

5{tex} m v _ { 0 } ^ { 2 } {/tex}

D

6{tex} m v _ { 0 } ^ { 2 } {/tex}

Explanation

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Q 9. Consider a uniform square plate of side {tex} ^ { \prime } a ^ { \prime } {/tex} and mass {tex} ^ { \prime } M ^ { \prime } {/tex} . The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

A

{tex} \frac { 5 } { 6 } M a ^ { 2 } {/tex}

B

{tex} \frac { 1 } { 12 } M a ^ { 2 } {/tex}

C

{tex} \frac { 7 } { 12 } M a ^ { 2 } {/tex}

{tex} \frac { 2 } { 3 } M a ^ { 2 } {/tex}

Explanation





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Q 10. A dancer is standing on a stool rotating about the vertical axis passing through its centre. She pulls her arms towards the body reducing her moment of inertia by a factor of n. The new angular speed of turn table is proportional to

n

B

{tex} n ^ { - 1 } {/tex}

C

{tex} n ^ { 0 } {/tex}

D

{tex} n ^ { 2 } {/tex}

Explanation

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Q 11. A uniform square plate has a small piece {tex} Q {/tex} of an irregular shape removed and glued to the centre of the plate leaving a hole behind. Then the moment of inertia about the Z-axis

A

increases

decreases

C

remains same

D

changed in unpredicted manner.

Explanation

In the given diagram, when the small piece Q removed and glued to the centre of the plate, the mass comes closer to the z-axis, hence moment of interia decreases.

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Q 12. A circular turn table has a block of ice placed at its centre. The system rotates with an angular speed {tex} \omega {/tex} about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system

A

becomes zero

B

remains constant at the same value {tex} \omega {/tex}

C

increases to a value greater than {tex} \omega {/tex}

decreases to a value less than {tex} \omega {/tex}

Explanation

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Q 13. Seven identical coins are rigidly arranged on a flat table in the pattern shown below so that each coin touches it neighbors. Each coin is a thin disc of mass {tex} m {/tex} and radius {tex} r . {/tex} The moment of inertia of the system of seven coins about an axis that passes through point {tex} P {/tex} and perpendicular to the plane of the coin is:

A

{tex} \frac { 55 } { 2 } m r ^ { 2 } {/tex}

B

{tex} \frac { 127 } { 2 } m r ^ { 2 } {/tex}

{tex} \frac { 111 } { 2 } m r ^ { 2 } {/tex}

D

{tex}55 { mr } ^ { 2 } {/tex}

Explanation

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Q 14. In a two-particle system with particle masses {tex} \mathrm { m } _ { 1 } {/tex} and {tex} \mathrm { m } _ { 2 } , {/tex} the first particle is pushed towards the centre of mass through a distance {tex} \mathrm { d }{/tex}, the distance through which second particle must be moved to keep the centre of mass at the same position is

A

{tex} \frac { \mathrm { m } _ { 2 } \mathrm { d } } { \mathrm { m } _ { 1 } } {/tex}

B

{tex} \mathrm { d } {/tex}

C

{tex} \frac { \mathrm { m } _ { 1 } \mathrm { d } } { \left( \mathrm { m } _ { 1 } + \mathrm { m } _ { 2 } \right) } {/tex}

{tex} \frac { \mathrm { m } _ { 1 } \mathrm { d } } { \mathrm { m } _ { 2 } } {/tex}

Explanation


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Q 15. A uniform bar of mass {tex} \mathrm { M } {/tex} and length {tex} \mathrm { L } {/tex} is horizontally suspended from the ceiling by two vertical light cables as shown. Cable {tex} \mathrm { A } {/tex} is connected 1⁄4 distance from the left end of the bar. Cable {tex} \mathrm { B } {/tex} is attached at the far right end of the bar. What is the tension in cable {tex} \mathrm { A } {/tex}?

A

{tex}1 / 4 \mathrm { Mg } {/tex}

B

{tex}1 / 3 \mathrm { Mg } {/tex}

{tex} 2/ 3 \mathrm { Mg } {/tex}

D

{tex} 3/ 4 \mathrm { Mg } {/tex}

Explanation



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Q 16. A couple produces

A

purely linear motion

purely rotational motion

C

linear and rotational motion

D

no motion

Explanation

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Q 17. Point masses {tex} 1,2,3 {/tex} and 4{tex} \mathrm { kg } {/tex} are lying at the point {tex} ( 0,0,0 ) , {/tex} {tex} ( 2,0,0 ) , ( 0,3,0 ) {/tex} and {tex} ( - 2 , - 2,0 ) {/tex} respectively. The moment of inertia of this system about {tex} \mathrm { x } {/tex} -axis will be

43{tex} \mathrm { kgm } ^ { 2 } {/tex}

B

34{tex} \mathrm { kgm } ^ { 2 } {/tex}

C

27{tex} \mathrm { kgm } ^ { 2 } {/tex}

D

72{tex} \mathrm { kgm } ^ { 2 } {/tex}

Explanation



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Q 18. A solid sphere of mass M and radius R is pulled horizontally on a sufficiently rough surface as shown in the figure. Choose the correct alternative.

A

The acceleration of the centre of mass is {tex} \frac {F}{M} {/tex}

The acceleration of the centre of mass is {tex} \frac {2F}{3M} {/tex}

C

The friction force on the sphere acts forward

D

The magnitude of the friction force is {tex} \frac {F}{3} {/tex}

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Q 19. The moment of inertia of a body about a given axis is 1.2 kg m {tex} ^ { 2 } {/tex}. Initially, the body is at rest. In order to produce a rotational kinetic energy of 1500 joule, an angular acceleration of 25 radian/sec{tex} ^ { 2 } {/tex} must be applied about that axis for a duration of

A

4 sec

2 sec

C

8 sec

D

10 sec

Explanation

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Q 20. A gymnast takes turns with her arms and legs stretched. When she pulls her arms and legs in

A

the angular velocity decreases

the moment of inertia decreases

C

the angular velocity stays constant

D

the angular momentum increases

Explanation

Since no external torque act on gymnast, so angular momentum ( L = Iω ) is conserved. After pulling her arms & legs, the angular velocity increases but moment of inertia of gymnast, decreases in, such a way that angular momentum remains constant.

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Q 21. An equilateral triangle {tex} A B C {/tex} formed from a uniform wire has two small identical beads initially located at {tex} A {/tex} . The triangle is set rotating about the vertical axis {tex} A O {/tex} . Then the beads are released from rest simultaneously and allowed to slide down, one along {tex} A B {/tex} and the other along {tex} A C {/tex} as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are

A

angular velocity and total energy (kinetic and potential)

total angular momentum and total energy

C

angular velocity and moment of inertia about the axis of rotation

D

total angular momentum and moment of inertia about the axis of rotation

Explanation

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Q 22. The moment of inertia of a uniform semicircular wire of mass {tex} \mathrm { m } {/tex} and radius {tex} \mathrm { r } , {/tex} about an axis passing through its centre of mass and perpendicular to its plane is {tex} \mathrm { mr } ^ { 2 } \left( 1 - \frac { \mathrm { k } } { \pi ^ { 2 } } \right) . {/tex} Find the value of {tex} \mathrm { k } {/tex} .

A

2

B

3

4

D

5

Explanation



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Q 23. Initial angular velocity of a circular disc of mass {tex} M {/tex} is {tex} \omega _ { 1 } {/tex} Then two small spheres of mass {tex} m {/tex} are attached gently to
diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

A

{tex} \left( \frac { M + m } { M } \right) \omega _ { 1 } {/tex}

B

{tex} \left( \frac { M + m } { m } \right) \omega _ { 1 } {/tex}

{tex} \left( \frac { M } { M + 4 m } \right) \omega _ { 1 } {/tex}

D

{tex} \left( \frac { M } { M + 2 m } \right) \omega _ { 1 } {/tex}

Explanation





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Q 24. Five masses are placed in a plane as shown in figure. The coordinates of the centre of mass are nearest to

A

{tex} 1.2,1.4 {/tex}

B

{tex} 1.3,1.1 {/tex}

{tex} 1.1,1.3 {/tex}

D

{tex} 1.0,1.0 {/tex}

Explanation