### JEE Mains Previous Years Questions with Solutions

4.5
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1

### AIEEE 2007

A circular disc of radius $R$ is removed from a bigger circular disc of radius $2R$ such that the circumferences of the discs coincide. The center of mass of the new disc is $\alpha R$ form the center of the bigger disc. The value of $\alpha$ is
A
$1/4$
B
$1/3$
C
$1/2$
D
$1/6$

## Explanation

Let the mass per unit area be $\sigma .$

Then the mass of the complete disc
$= \sigma \left[ {\pi {{\left( {2R} \right)}^2}} \right] = 4\pi \sigma {R^2}$

The mass of the removed disc $= \sigma \left( {\pi {R^2}} \right) = \pi \sigma {R^2}$

So mass of the remaining disc = $4\pi \sigma {R^2}$ - $\pi \sigma {R^2}$ = $3\pi \sigma {R^2}$

Let center of mass of $3\pi \sigma {R^2}$ mass is at x distance from origin O.

$\therefore$ ${{3\pi {R^2}\sigma .x + \pi {R^2}\sigma .R} \over {4\pi {R^2}\sigma }} = 0$

As center of mass of full disc is at Origin.

$\therefore$ $x = - {R \over 3}$

According to the question, $x$ = $\alpha R$

$\therefore$ $\alpha = - {1 \over 3}$

$\Rightarrow$ $\left| \alpha \right| = {1 \over 3}$

2

### AIEEE 2006

A force of $- F\widehat k$ acts on $O,$ the origin of the coordinate system. The torque about the point $(1, -1)$ is
A
$F\left( {\widehat i - \widehat j} \right)$
B
$- F\left( {\widehat i + \widehat j} \right)$
C
$F\left( {\widehat i + \widehat j} \right)$
D
$- F\left( {\widehat i - \widehat j} \right)$

## Explanation

We know, Torque $\overrightarrow \tau = \overrightarrow r \times \overrightarrow F$

$= \left( {\widehat i - \widehat j} \right) \times \left( { - F\widehat k} \right)$

$= F\left( {\widehat i + \widehat j} \right)$
3

### AIEEE 2006

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega ' =$
A
${{\omega \left( {m + 2M} \right)} \over m}$
B
${{\omega \left( {m - 2M} \right)} \over {\left( {m + 2M} \right)}}$
C
${{\omega m} \over {\left( {m + M} \right)}}$
D
${{\omega m} \over {\left( {m + 2M} \right)}}$

## Explanation

Here angular momentum is conserved.

Applying conservation of angular momentum $I'\omega ' = I\omega \,\,$

$\left( {m{R^2} + 2M{R^2}} \right)\omega \,' = m{R^2}\omega$

$\Rightarrow \omega \,' = \omega \left[ {{m \over {m + 2M}}} \right]$
4

### AIEEE 2006

Four point masses, each of value $m,$ are placed at the corners of a square $ABCD$ of side $l$. The moment of inertia of this system about an axis passing through $A$ and parallel to $BD$ is
A
$2m{l^2}$
B
$\sqrt 3 m{l^2}$
C
$3m{l^2}$
D
$m{l^2}$

## Explanation

Let ${I_{A}}$ is the moment of inertia about an axis passing through A and parallel to BD.

${I_{A}} = M.I\,$ due to the point mass at $B+$

$M.I$ due to the point mass at $D+$

$M.I$ due to the point mass at $C.$

${I_{A}} = 2 \times m{\left( {{\textstyle{\ell \over {\sqrt 2 }}}} \right)^2} + m{\left( {\sqrt 2 \ell } \right)^2}$

$= m{\ell ^2} + 2m{\ell ^2} = 3m{\ell ^2}$