 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2005

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the center is
A
${2 \over 5}M{r^2}$
B
${1 \over 4}Mr$
C
${1 \over 2}M{r^2}$
D
$M{r^2}$

Explanation

Let mass of the semi circular disc = M

Now assume a disc which is combination of two semi circular parts. Let $I$ be the moment of inertia of the uniform semicircular disc. So $2I$ will be the moment of inertia of the full circular disc and 2M will be the mass.

$\Rightarrow 2I = {{2M{r^2}} \over 2}$

$\Rightarrow I = {{M{r^2}} \over 2}$
2

AIEEE 2005

An annular ring with inner and outer radii ${R_1}$ and ${R_2}$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, ${{{F_1}} \over {{F_2}}}\,$ is
A
${\left( {{{{R_1}} \over {{R_2}}}} \right)^2}$
B
${{{{R_2}} \over {{R_1}}}}$
C
${{{{R_1}} \over {{R_2}}}}$
D
$1$

Explanation

Let the mass of each particle is m.

Then force experienced by each particle, $F = m{\omega ^2}R$

$\therefore$ ${{{F_1}} \over {{F_2}}} = {{m{\omega ^2}{R_1}} \over {m{\omega ^2}{R_2}}}$

$\Rightarrow$ ${{{F_1}} \over {{F_2}}} = {{{R_1}} \over {{R_2}}}$
3

AIEEE 2004

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which on of the following will not be affected ?
A
Angular velocity
B
Angular momentum
C
Moment of inertia
D
Rotational kinetic energy

Explanation

Solid sphere is rotating in free space that means no external torque is operating on the sphere.

Angular momentum will remain the same since external torque is zero.
4

AIEEE 2004

One solid sphere $A$ and another hollow sphere $B$ are of same mass and same outer radii. Their moment of inertia about their diameters are respectively ${I_A}$ and ${I_B}$ such that
A
${I_A} < {I_B}$
B
${I_A} > {I_B}$
C
${I_A} = {I_B}$
D
${{{I_A}} \over {{I_B}}} = {{{d_A}} \over {{d_B}}}$ where ${d_A}$ and ${d_B}$ are their densities.

Explanation

For solid sphere the moment of inertia of $A$ about its diameter

${I_A} = {2 \over 5}M{R^2}.$

The moment of inertia of a hollow sphere $B$ about its diameter

${I_B} = {2 \over 3}M{R^2}.$

$\therefore$ ${I_A} < {I_B}$