Moment of Inertia

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JEE Main 2019, 11 Jan Shift-II

A circular disc $D_{1}$ of mass $M$ and radius $R$ has two identical discs $D_{2}$ and $D_{3}$ of the same mass $M$ and radius $R$ attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis $O O^{'}$ passing through the centre of $D_{1}$ , as shown in the figure will be

\frac{2}{3} M R^{2}

(B)

\frac{4}{5} M R^{2}

(C)

3 M R^{2}

(D)

M R^{2}

JEE Main 2019, 10 April Shift-II

LEVELJEE Main

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The first part has a mass of $7 M /8$ and is converted into a uniform disc of radius $2 R$ . The second part is converted into a uniform solid sphere. Let $I_{1}$ be the moment of inertia of the disc about its axis and $I_{2}$ be the moment of inertia of the new sphere about its axis. The ratio $I_{1} / I_{2}$ is given by

JEE Main 2020, 7 Jan Shift-II

LEVELJEE Main

Mass per unit area of a circular disc of radius $a$ depends on the distance $r$ from its centre as $σ (r) = A + B r$ . The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is

(A)

2 π a^{4} (\frac{A}{4} + \frac{B}{5})

(B)

2 π a^{4} (\frac{a A}{4} + \frac{B}{5})

(C)

π a^{4} (\frac{A}{4} + \frac{a B}{5})

(D)

2 π a^{4} (\frac{A}{4} + \frac{a B}{5})

LEVELJEE Advanced

For the given uniform square lamina $A B C D$ , whose centre is $O$

(A)

2 I_{A C} = I_{E F}

(B)

I_{A D} = 3 I_{E F}

(C)

I_{A D} = 4 I_{E F}

(D)

I_{A C} = 2 I_{E F}

JEE Main 2020, 06 Sep Shift-I

LEVELJEE Main

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is $M$ , radius of its top $R$ and height $H$ , then its moment of inertia about its axis is

(A)

\frac{M R ^{2}}{2}

(B)

\frac{M ( R ^{2} + H ^{2} )}{4}

(C)

\frac{M H ^{2}}{3}

(D)

\frac{M R ^{2}}{3}

JEE Main 2021, 31 Aug Shift-II

LEVELJEE Main

A system consists of two identical spheres each of mass $1.5 kg$ and radius $50 cm$ at the end of light rod. The distance between the centres of the two spheres is $5 m$ . What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its mid-point ?

(A)

18.75 kgm^{2}

(B)

1.905 \times 1 0^{5} kgm^{2}

(C)

19.05 kgm^{2}

(D)

1.875 \times 1 0^{5} kgm^{2}

LEVELJEE Main

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)

\frac{I _{A}}{d _{A}} = \frac{I _{B}}{d _{B}}

where

d_{A}

and

d_{B}

are their densities.

JEE Main 2019, 12 April Shift-I

LEVELJEE Advanced

A circular disc of radius $b$ has a hole of radius $a$ at its centre (see figure). If the mass per unit area of the disc varies as $(\frac{σ _{0}}{r})$ , then the radius of gyration of the disc about its axis passing through the centre is

(A)

\frac{a ^{2} + b ^{2} + ab}{2}

(B)

\frac{a + b}{2}

(C)

\frac{a ^{2} + b ^{2} + ab}{3}

(D)

\frac{a + b}{3}

JEE Main 2021, 18 March Shift-II

LEVELJEE Main

Consider a uniform wire of mass $M$ and length $L$ . It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is

(A)

\frac{1}{4} \frac{M L ^{2}}{π ^{2}}

(B)

\frac{2}{5} \frac{M L ^{2}}{π ^{2}}

(C)

\frac{M L ^{2}}{π ^{2}}

(D)

\frac{1}{2} \frac{M L ^{2}}{π ^{2}}

LEVELJEE Main

Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is

(A)

\frac{M R ^{2}}{2}

(B)

M R^{2}

(C)

2 M R^{2}

(D)

\frac{M R ^{2}}{4}

JEE Main 2019

LEVELJEE Main

An equilateral triangle $A B C$ is cut from a thin solid sheet of wood. (see figure) $D, E$ and $F$ are the mid points of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_{0}$ . If the smaller triangle $D E F$ is removed from $A B C$ , the moment of inertia of the remaining figure about the same axis is $I$ . Then

(A)

I^{'} = \frac{3}{4} I_{0}

(B)

I^{'} = \frac{15}{16} I_{0}

(C)

I^{'} = \frac{I _{0}}{4}

(D)

I^{'} = \frac{9}{16} I_{0}

JEE Main 2021, 25 July Shift-1

LEVELJEE Main

Assertion A: Moment of inertia of a circular disc of mass $M$ and radius $R$ about $X, Y$ -axes (passing through its plane) and $Z$ -axis which is perpendicular to its plane were found to be $I_{x}, I_{y}$ and $I_{z}$ , respectively. The respective radii of gyration about all the three axes will be the same. Reason R: A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below.

(A)

Both A and R are correct and R is the correct explanation of A.

(B)

Both A and R are correct but R is not the correct explanation of A.

(C)

A is correct but R is not correct.

(D)

A is not correct but R is correct.

JEE Main 2018

LEVELJEE Advanced

Seven identical circular planar discs, each of mass $M$ and radius $R$ are welded symmetrically as shown in the figure. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point $P$ is

(A)

\frac{19}{2} M R^{2}

(B)

\frac{55}{2} M R^{2}

(C)

\frac{73}{2} M R^{2}

(D)

\frac{181}{2} M R^{2}

JEE Main 2021, 24 Feb Shift-II

LEVELJEE Advanced

A uniform thin bar of mass 6 kg and length 2.4 m is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is ……… $\times 1 0^{- 1} kg -m^{2}$ .

JEE Main 2020

LEVELJEE Advanced

Three solid sphere each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$ . The ratio $I_{0} / I_{A}$ of moment of inertia $I_{0}$ of the system about an axis passing the centroid and about centre of any of the spheres $I_{A}$ and perpendicular to the plane of the triangle is

(A)

\frac{15}{13}

(B)

\frac{13}{15}

(C)

\frac{13}{23}

(D)

\frac{23}{13}

JEE Main 2019, 8 April Shift-I

LEVELJEE Advanced

A thin circular plate of mass $M$ and radius $R$ has its density varying as $ρ (r) = ρ_{0} r$ with $ρ_{0}$ as constant and $r$ is the distance from its centre. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = a M R^{2}$ . The value of the coefficient $a$ is

(A)

\frac{1}{2}

(B)

\frac{3}{5}

(C)

\frac{8}{5}

(D)

\frac{3}{2}

JEE Main 2020

LEVELJEE Main

Four point masses, each of mass $m$ are fixed at the corners of a square of side $l$ . The square is rotating with angular frequency $ω$ , about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is

(A)

m l^{2} ω

(B)

4 m l^{2} ω

(C)

3 m l^{2} ω

(D)

2 m l^{2} ω

JEE Main 2019, 12 Jan Shift-II

LEVELJEE Main

The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of $x$ from it, is ' $I (x)$ '. Which one of the graphs represents the variation of $I (x)$ with $x$ correctly?

Consider a uniform square plate of side $a$ and mass $m$ . The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

(A)

\frac{5}{6} m a^{2}

(B)

\frac{1}{12} m a^{2}

(C)

\frac{7}{12} m a^{2}

(D)

\frac{2}{3} m a^{2}

JEE Main 2019

LEVELJEE Main

Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2 R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

(A)

\frac{137}{15} M R^{2}

(B)

\frac{209}{15} M R^{2}

(C)

\frac{17}{15} M R^{2}

(D)

\frac{152}{15} M R^{2}

JEE Main 2021, 26 Aug Shift-I

LEVELJEE Advanced

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same ( $M$ ) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, $\frac{r}{2}$ distance from the end $A$ of the handle will be ........ $M r^{2}$ .

JEE Main 2019, 10 April Shift-I

LEVELJEE Main

A thin disc of mass $M$ and radius $R$ has mass per unit area $σ (r) = k r^{2}$ , where $r$ is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is

(A)

\frac{M R ^{2}}{2}

(B)

\frac{M R ^{2}}{6}

(C)

\frac{M R ^{2}}{3}

(D)

\frac{2 M R ^{2}}{3}

LEVELJEE Main

A circular disc $X$ of radius $R$ is made from an iron plate of thickness $t$ and another disc $Y$ of radius $4 R$ is made from an iron plate of thickness $t /4$ . Then, the relation between the moment of inertia $I_{X}$ and $I_{Y}$ is

(A)

I_{Y} = 32 I_{X}

(B)

I_{Y} = 16 I_{X}

(C)

I_{Y} = I_{X}

(D)

I_{Y} = 64 I_{X}

JEE Main 2021, 27 July Shift-I

LEVELJEE Main

The figure shows two solid discs with radius $R$ and $r$ , respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis $A B$ (which is perpendicular to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane?

(A)

R^{2} : r^{2}

(B)

2 r^{4} : R^{4}

(C)

2 R^{2} : r^{2}

(D)

2 R^{4} : r^{4}

JEE Main 2020 (06 Sep Shift-II)

LEVELJEE Advanced

The linear mass density of a thin rod $A B$ of length $L$ varies from $A$ to $B$ as $λ (x) = λ_{0} (1 + \frac{x}{L})$ , where $x$ is the distance from $A$ . If $M$ is mass of the rod, then its moment of inertia about an axis passing through $A$ and perpendicular to the rod is

(A)

\frac{5}{12} M L^{2}

(B)

\frac{7}{18} M L^{2}

(C)

\frac{2}{5} M L^{2}

(D)

\frac{3}{7} M L^{2}

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M, radius of its top R and height H, then its moment of inertia about its axis is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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 IA​ and IB​ such that

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Moment of inertia of a circular wire of mass M and radius R about its diameter is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider a uniform square plate of side a and mass m. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

For the given uniform square lamina $A B C D$ , whose centre is $O$

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is $M$ , radius of its top $R$ and height $H$ , then its moment of inertia about its axis is

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

Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is

Consider a uniform square plate of side $a$ and mass $m$ . The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is