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Q 1.

Correct4

Incorrect-1

A long horizontal rod has a bead which can slide along its length and initially placed at a distance {tex} L {/tex} from one end {tex} A {/tex} of the rod. The rod is set in angular motion about {tex} A {/tex} with constant angular acceleration {tex} \alpha . {/tex} If the coefficient of friction between the rod and the bead is {tex} \mu , {/tex} and gravity is neglected, then the time after which the bead starts slipping is

{tex} \sqrt { \mu / \alpha } {/tex}

{tex} \mu / \sqrt { \alpha } {/tex}

{tex} \frac { 1 } { \sqrt { \mu \alpha } } {/tex}

infinitesimal

Q 2.

Correct4

Incorrect-1

A cubical block of side {tex} L {/tex} rests on a rough horizontal surface with coefficient of friction {tex} \mu {/tex}. A horizontal force {tex} F {/tex} is applied on the block as shown. If the coefficient of friction is sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is

infinitesimal

{tex} \mathrm { mg } / 4 {/tex}

{tex} \mathrm { mg } / 2 {/tex}

{tex} \mathrm { mg } ( 1 - \mu ) {/tex}

Q 3.

Correct4

Incorrect-1

A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved?

centre of the circle

on the circumference of the circle.

inside the circle

outside the circle.

Q 4.

Correct4

Incorrect-1

A particle is confined to rotate in a circular path decreasing linear speed, then which of the following is correct?

{tex} \vec { L } {/tex} (angular momentum) is conserved about the centre

only direction of angular momentum {tex} \vec { L } {/tex} is conserved

It spirals towards the centre

its acceleration is towards the centre.

Q 5.

Correct4

Incorrect-1

A bob of mass {tex} M {/tex} is suspended by a massless string of length {tex} L . {/tex} The horizontal velocity {tex} v {/tex} at position {tex} A {/tex} is just sufficient to make it reach the point {tex} B . {/tex} The angle {tex} \theta {/tex} at which the speed of the bob is half of that at {tex}A {/tex}, satisfies

{tex} \theta = \frac { \pi } { 4 } {/tex}

{tex} \frac { \pi } { 4 } < \theta < \frac { \pi } { 2 } {/tex}

{tex} \frac { \pi } { 2 } < \theta < \frac { 3 \pi } { 4 } {/tex}

{tex} \frac { 3 \pi } { 4 } < \theta < \pi {/tex}

Q 6.

Correct4

Incorrect-1

A small mass {tex} m {/tex} is attached to a massless string whose other end is fixed at {tex} P {/tex} as shown in the figure. The mass is undergoing circular motion in the {tex} x - y {/tex} plane with centre at {tex} O {/tex} and constant angular speed {tex} \omega . {/tex} If the angular momentum of the system, calculated about {tex} O {/tex} and {tex} P {/tex} are denoted by {tex} \vec { L } _ { O } {/tex} and {tex} \vec { L } _ { P } {/tex} respectively, then

{tex} \vec { L } _ { O } {/tex} and {tex} \vec { L } _ { P } {/tex} do not vary with time

{tex} \vec { L } _ { O } {/tex} varies with time while {tex} \vec { L } _ { P } {/tex} remains constant

{tex} \vec { L } _ { O } {/tex} remains constant while {tex} \vec { L } _ { P } {/tex} varies with time

{tex} \vec { L } _ { O } {/tex} and {tex} \vec { L } _ { P } {/tex} both vary with time

Q 7.

Correct4

Incorrect-1

A thin uniform rod, pivoted at {tex} O , {/tex} is rotating in the horizontal plane with constant angular speed {tex} \omega , {/tex} as shown in the figure. At time {tex} t = 0 , {/tex} a small insect starts from {tex} O {/tex} and moves with constant speed {tex} v , {/tex} with respect to the rod towards the other end. It reaches the end of the rod at {tex} t = T {/tex} and stops. The angular speed of the system remains {tex}\omega{/tex} throughout. The magnitude of the torque {tex} ( | \vec { \tau } | ) {/tex} about {tex} O , {/tex} as a function of time is best represented by which plot?

Q 8.

Correct4

Incorrect-1

Consider regular polygons with number of sides {tex} n{/tex} = 3, 4 , 5.... as shown in the figure. The center of mass of all the polygons is at height {tex}h{/tex} from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is {tex} \Delta . {/tex} Then {tex} \Delta {/tex} depends on {tex} n{/tex} and {tex}h{/tex} as

{tex} \Delta = { h } \sin ^ { 2 } \left( \frac { \pi } { { n } } \right) {/tex}

{tex} \Delta = { h } \left( \frac { 1 } { \cos \left( \frac { \pi } { { n } } \right) } - 1 \right) {/tex}

{tex} \Delta = { h } \sin \left( \frac { 2 \pi } { { n } } \right) {/tex}

{tex} \Delta = { h } \tan ^ { 2 } \left( \frac { \pi } { 2 { n } } \right) {/tex}

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