1

### JEE Main 2019 (Online) 9th January Morning Slot

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is $\theta$, thrown : A
tan$\theta$ = ${1 \over {2\sqrt 3 }}$
B
tan$\theta$ = ${1 \over 2}$
C
tan$\theta$ = ${2 \over {\sqrt 3 }}$
D
tan$\theta$ = ${1 \over 3}$

## Explanation Assume mass of each part = m

Here, ${{{C_1}R} \over {{C_1}A}} = \sin \theta$

$\therefore$   C1R = C1A(sin$\theta$)

= ${1 \over 2}$ sin$\theta$

${{BY} \over {AB}} = \sin \theta$

$\Rightarrow$   BY = Lsin$\theta$

As By = MP

$\therefore$   MP = Lsin$\theta$

${{{C_2}M} \over {B{C_2}}}$ = cos$\theta$

$\Rightarrow$   C2M = BC2cos$\theta$

$\Rightarrow$   C2M = ${L \over 2}\cos \theta$

$\therefore$   C2P = C2M $-$ MP = ${L \over 2}\cos \theta$ $-$ Lsin$\theta$

Now balancing torque about hinge paint A,

mg(C1R) = mg(C2P)

$\Rightarrow$   mg$\left( {{L \over 2}\sin \theta } \right)$ = mg$\left( {{L \over 2}\cos \theta - L\sin \theta } \right)$

$\Rightarrow$   ${{\sin \theta } \over 2}$ = ${{\cos \theta } \over 2} -$ sin$\theta$

$\Rightarrow$   ${{3\sin \theta } \over 2}$ = ${{\cos \theta } \over 2}$

$\Rightarrow$   tan$\theta$ = ${1 \over 3}$
2

### JEE Main 2019 (Online) 9th January Morning Slot

If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is :
A
${L \over m}$
B
${4L \over m}$
C
${L \over 2m}$
D
${2L \over m}$

## Explanation dA = ${1 \over 2}$ r2d$\theta$

$\therefore$   ${{dA} \over {dt}} = {1 \over 2}{r^2}{{d\theta } \over {dt}}$

$\Rightarrow$   ${{dA} \over {dt}} = {1 \over 2}{r^2}\omega$    . . . . . (1)

We know,

angular momentum,

L = $mvr$

= $m\left( {\omega r} \right)r$

= mr2$\omega$

$\therefore$   $\omega$ = ${L \over {m{r^2}}}$     . . . . . (2)

Put value of $\omega$ in equation(1),

${{dA} \over {dt}}$ = ${1 \over 2}{r^2}$ (${L \over {m{r^2}}}$)

= ${L \over {2m}}$
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### JEE Main 2019 (Online) 9th January Evening Slot

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30o from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad s$-$1) will be (g = 10 ms$-$2) A
$\sqrt {{{30} \over 2}}$
B
$\sqrt {30}$
C
${{\sqrt {20} } \over 3}$
D
${{\sqrt {30} } \over 2}$

## Explanation When this rod move from initial position to final position then,

Gain in kinetic energy = loss in potential energy

## Explanation Consider a strip of radius x & thickness dx,

Torque due to friction on this strip.

$\int {d\tau = \int\limits_0^R {{{x\mu F.2\pi xdx} \over {\pi {R^2}}}} }$

$\tau = {{2\mu F} \over {{R^2}}}.{{{R^3}} \over 3}$

$\tau = {{2\mu FR} \over 3}$