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### JEE Main 2017 (Online) 8th April Morning Slot

Moment of inertia of an equilateral triangular lamina ABC, about the axis passing through its centre O and perpendicular to its plane is Io as shown in the figure. A cavity DEF is cut out from the lamina, where D, E, F are the mid points of the sides. Moment of inertia of the remaining part of lamina about the same axis is :

A
${7 \over 8}$ Io
B
${15 \over 16}$ Io
C
${{3\,{{\rm I}_o}} \over 4}$
D
${{31\,{{\rm I}_o}} \over 32}$

## Explanation

Let, side of triangle ABC = $\ell$

According to perpendicular axes theorem, moment of inertia of triangle about it center and perpendicular to its plane,

IO = ${1 \over {12}}$ m$\ell$2

In, triangle DEF,

DE = DF = EF = ${1 \over 2}$ AB = ${1 \over 2}$ $\ell$

$\therefore\,\,\,$ moment of inertia of triangle DEF,

IDEF = ${1 \over {12}} \times {m \over 4} \times {\left( {{\ell \over 2}} \right)^2}$

= ${1 \over {12}} \times {{m{\ell ^2}} \over {16}}$

= ${{{I_O}} \over {16}}$

$\therefore\,\,\,$ Moment of inertia of the remaining part,

Iremain = IO $-$ ${{{{\rm I}_O}} \over {16}}$ = ${{15\,{{\rm I}_O}} \over {16}}$
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### JEE Main 2017 (Online) 8th April Morning Slot

In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?
A
Left arm is longer than the right arm
B
Both the arms are of same length
C
Left arm is shorter than the right arm
D
Every object that is weighed using this balance appears lighter than its actual weight.

## Explanation

From principle of moment we know, the anticlockwise moment is equal to clockwise moment when a system is stable or balance.

$\therefore\,\,\,$ load $\times$ load arm = effect $\times$ effect arm

When 5 mg weight is placed on the left pan, load arm shift to left side, hence left arm become shorter than right arm.
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### JEE Main 2017 (Online) 8th April Morning Slot

A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is :
A
${{2\,\,mg} \over {2\,m + M}}$
B
${{2\,\,Mg} \over {2\,m + M}}$
C
${{2\,\,mg} \over {2\,M + m}}$
D
${{2\,\,Mg} \over {2\,M + M}}$

## Explanation

From figure, we can say

ma = mg $-$ T . . . . . (1)

Moment of inertia of a uniform disc,

I = ${1 \over 2}$ MR2

and the acceleration of the disc, a = $\propto$ R

We know,

Torque = I$\propto$ = RT

$\therefore\,\,\,$ RT = ${1 \over 2}$ M R2 $\times$ ${a \over R}$ = ${{MaR} \over 2}$

$\Rightarrow$ $\,\,\,$ T = ${{Ma} \over 2}$

Putting the value of T in eq (1).

ma = mg $-$ ${{Ma} \over 2}$

$\Rightarrow$ $\,\,\,$ a ( m + ${{Ma} \over 2}$) = mg

$\Rightarrow$ $\,\,\,$ a = ${{2mg} \over {2m + M}}$
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### JEE Main 2017 (Online) 9th April Morning Slot

The machine as shown has 2 rods of length1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :

A
Constant speed
B
decreasing speed
C
increasing speed
D
speed which is ${3 \over 4}$th of that of the roller when the weight is 0.4 m above the ground