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Calculate the Moment of Inertia of an annular disc of mass $M$ and inner radius $R_{1}$ and outer radius $R_{2}$ about an axis tangential and perpendicular to the plane.

Important Questions on Rotation

HARD

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

From a uniform circular disc of radius R and mass 9 M, a small disc of radius

\frac{R}{3}

is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:

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MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

A mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

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HARD

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

When a

12000 J

of work is done on a flywheel, its frequency of rotation increases from

10 Hz to 20 Hz

. The moment of inertia of flywheel about its axis of rotation is

(π^{2} = 10)

EASY

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

The moment of inertia of a uniform cylinder of length

l

and radius

R

about its perpendicular bisector is

I .

What is the ratio

l / R

such that the moment of inertia is minimum?

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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 inertia about an axis going through its centre of mass and perpendicular to its plane is:

EASY

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

Consider a thin uniform square sheet made of a rigid material. If its side is

a

, mass m and moment of inertia

I

about one of its diagonals, then:

HARD

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

Moment of inertia of a cylinder of mass

m,

length

L

and radius

R

about an axis passing through its centre and perpendicular to the axis of the cylinder is

I = M (\frac{R^{2}}{4} + \frac{L^{2}}{12}) .

If such a cylinder is to be made for a given mass of a material, the ratio

\frac{L}{R}

for it to have minimum possible

I

is:

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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

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MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

From a circular cardboard of uniform thickness and mass

M

, a square disc of the maximum possible area is cut. If the moment of inertia of the square with the axis of rotation at the centre and perpendicular to the plane of the disc is

\frac{M a^{2}}{6},

the radius of the circular cardboard is

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

Three identical rods each of mass

M

and length

L

are joined to form a symbol

H

. The moment of inertia of the system about one of the sides of

H

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

From a disc of radius

R

and mass

M

, a circular hole of diameter

R

, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?

HARD

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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:

HARD

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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:

EASY

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

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For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through $O$ (the centre of mass) and $O$ ' (corner point) is:

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

A circular disc of mass

M

and radius

R

is rotating about its axis with angular speed

ω_{1}

. If another stationary disc having radius

\frac{R}{2}

and same mass

M

is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed

ω_{2}

. The energy lost in the process is

p %

of the initial energy. Value of

p

is _____.

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

The linear mass density of a thin rod

AB

of length

L

varies from

A

B

λ (x) = λ_{0} (1 + \frac{x}{L})

, where

x

is the distance from

A

. If

M

is the mass of the rod then its moment of inertia about an axis passing through

A

and perpendicular to the rod :

MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

A circular hole of radius

\frac{R}{4}

is made in a thin uniform disc having mass and radius

R

, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point

O

and perpendicular to the plane of the disc is-
Question Image

EASY

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

A thin circular disk is in the

x y

plane as shown in the figure. The ratio of its moment of inertia about

z

and

z'

axes will be:
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MEDIUM

Physics>Mechanics>Rotation>Calculation of Moment of Inertia

From a solid sphere of mass

M

and radius

R

, a cube of the maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is:

Calculate the Moment of Inertia of an annular disc of mass M and inner radius R1 and outer radius R2about an axis tangential and perpendicular to the plane.

Important Questions on Rotation

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Calculate the Moment of Inertia of an annular disc of mass $M$ and inner radius $R_{1}$ and outer radius $R_{2}$ about an axis tangential and perpendicular to the plane.