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PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365930 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:
supporting img

1

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

2

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

3

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

4

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

Explanation:

supporting img
$I_{o} = I_{c m} + m d^{2}$
$= \frac{7 M R^{2}}{2} + 6 (M \times (2 R)^{2}) = \frac{55 M R^{2}}{2}$
$I_{P} = I_{o} + m d^{2}$
$= \frac{55 M R^{2}}{2} + 7 M (3 R^{2}) = \frac{181}{2} M R^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365931 Three identical thin rods, each of mass $m$ and length $ℓ$ , are joined to form an equilateral triangular frame. The moment of inertia of the frame about the axis parallel to its one side and passing through the opposite vertex is

1

\frac{3}{2} m ℓ^{2}

2

\frac{5}{4} m ℓ^{2}

3

\frac{5}{2} m ℓ^{2}

4

\frac{5}{3} m ℓ^{2}

Explanation:

Moment of inertia of each of $rod A C$ and $B C$ about the given axis $OO$ ' is
$I_{A C} = I_{B C} = \frac{m ℓ^{2}}{3} \sin^{2} 60^{\circ} = \frac{m ℓ^{2}}{4}$
supporting img
and $M . I$ of rod $A B$ about the given axis $O O$ is
$I_{A B} = m {(\frac{ℓ \sqrt{3}}{2})}^{2} = \frac{3}{4} m ℓ^{2}$
Hence,
$I = I_{A C} + I_{B C} + I_{A B} = \frac{m ℓ^{2}}{4} + \frac{m ℓ^{2}}{4} + \frac{3}{4} m ℓ^{2} = \frac{5}{4} m ℓ^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365932 Two masses each of mass $M$ are attached to the end of a rigid massless rod of length $L$ . The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is

1

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

2

2 M L^{2}

3

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

4

M L^{2}

Explanation:

The moment of inertia of the system about the given axis is
$I = M {(\frac{L}{2})}^{2} + M {(\frac{L}{2})}^{2} = \frac{M L^{2}}{4} + \frac{M L^{2}}{4} = \frac{M L^{2}}{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365933 Four thin identical rods are joined to form a square as shown. If it is rotated along an axis passing through one side, then its moment of inertia is equal to
supporting img

1

\frac{8}{3} m l^{2}

2

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

3

\frac{6}{3} m l^{2}

4

\frac{5}{3} m l^{2}

Explanation:

$I = \frac{m l^{2}}{3} + \frac{m l^{2}}{3} + m l^{2}$
$I = \frac{5}{3} m l^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365930 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:
supporting img

1

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

2

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

3

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

4

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

Explanation:

supporting img
$I_{o} = I_{c m} + m d^{2}$
$= \frac{7 M R^{2}}{2} + 6 (M \times (2 R)^{2}) = \frac{55 M R^{2}}{2}$
$I_{P} = I_{o} + m d^{2}$
$= \frac{55 M R^{2}}{2} + 7 M (3 R^{2}) = \frac{181}{2} M R^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365931 Three identical thin rods, each of mass $m$ and length $ℓ$ , are joined to form an equilateral triangular frame. The moment of inertia of the frame about the axis parallel to its one side and passing through the opposite vertex is

1

\frac{3}{2} m ℓ^{2}

2

\frac{5}{4} m ℓ^{2}

3

\frac{5}{2} m ℓ^{2}

4

\frac{5}{3} m ℓ^{2}

Explanation:

Moment of inertia of each of $rod A C$ and $B C$ about the given axis $OO$ ' is
$I_{A C} = I_{B C} = \frac{m ℓ^{2}}{3} \sin^{2} 60^{\circ} = \frac{m ℓ^{2}}{4}$
supporting img
and $M . I$ of rod $A B$ about the given axis $O O$ is
$I_{A B} = m {(\frac{ℓ \sqrt{3}}{2})}^{2} = \frac{3}{4} m ℓ^{2}$
Hence,
$I = I_{A C} + I_{B C} + I_{A B} = \frac{m ℓ^{2}}{4} + \frac{m ℓ^{2}}{4} + \frac{3}{4} m ℓ^{2} = \frac{5}{4} m ℓ^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365932 Two masses each of mass $M$ are attached to the end of a rigid massless rod of length $L$ . The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is

1

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

2

2 M L^{2}

3

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

4

M L^{2}

Explanation:

The moment of inertia of the system about the given axis is
$I = M {(\frac{L}{2})}^{2} + M {(\frac{L}{2})}^{2} = \frac{M L^{2}}{4} + \frac{M L^{2}}{4} = \frac{M L^{2}}{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365933 Four thin identical rods are joined to form a square as shown. If it is rotated along an axis passing through one side, then its moment of inertia is equal to
supporting img

1

\frac{8}{3} m l^{2}

2

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

3

\frac{6}{3} m l^{2}

4

\frac{5}{3} m l^{2}

Explanation:

$I = \frac{m l^{2}}{3} + \frac{m l^{2}}{3} + m l^{2}$
$I = \frac{5}{3} m l^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365930 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:
supporting img

1

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

2

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

3

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

4

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

Explanation:

supporting img
$I_{o} = I_{c m} + m d^{2}$
$= \frac{7 M R^{2}}{2} + 6 (M \times (2 R)^{2}) = \frac{55 M R^{2}}{2}$
$I_{P} = I_{o} + m d^{2}$
$= \frac{55 M R^{2}}{2} + 7 M (3 R^{2}) = \frac{181}{2} M R^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365931 Three identical thin rods, each of mass $m$ and length $ℓ$ , are joined to form an equilateral triangular frame. The moment of inertia of the frame about the axis parallel to its one side and passing through the opposite vertex is

1

\frac{3}{2} m ℓ^{2}

2

\frac{5}{4} m ℓ^{2}

3

\frac{5}{2} m ℓ^{2}

4

\frac{5}{3} m ℓ^{2}

Explanation:

Moment of inertia of each of $rod A C$ and $B C$ about the given axis $OO$ ' is
$I_{A C} = I_{B C} = \frac{m ℓ^{2}}{3} \sin^{2} 60^{\circ} = \frac{m ℓ^{2}}{4}$
supporting img
and $M . I$ of rod $A B$ about the given axis $O O$ is
$I_{A B} = m {(\frac{ℓ \sqrt{3}}{2})}^{2} = \frac{3}{4} m ℓ^{2}$
Hence,
$I = I_{A C} + I_{B C} + I_{A B} = \frac{m ℓ^{2}}{4} + \frac{m ℓ^{2}}{4} + \frac{3}{4} m ℓ^{2} = \frac{5}{4} m ℓ^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365932 Two masses each of mass $M$ are attached to the end of a rigid massless rod of length $L$ . The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is

1

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

2

2 M L^{2}

3

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

4

M L^{2}

Explanation:

The moment of inertia of the system about the given axis is
$I = M {(\frac{L}{2})}^{2} + M {(\frac{L}{2})}^{2} = \frac{M L^{2}}{4} + \frac{M L^{2}}{4} = \frac{M L^{2}}{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365933 Four thin identical rods are joined to form a square as shown. If it is rotated along an axis passing through one side, then its moment of inertia is equal to
supporting img

1

\frac{8}{3} m l^{2}

2

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

3

\frac{6}{3} m l^{2}

4

\frac{5}{3} m l^{2}

Explanation:

$I = \frac{m l^{2}}{3} + \frac{m l^{2}}{3} + m l^{2}$
$I = \frac{5}{3} m l^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365930 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:
supporting img

1

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

2

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

3

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

4

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

Explanation:

supporting img
$I_{o} = I_{c m} + m d^{2}$
$= \frac{7 M R^{2}}{2} + 6 (M \times (2 R)^{2}) = \frac{55 M R^{2}}{2}$
$I_{P} = I_{o} + m d^{2}$
$= \frac{55 M R^{2}}{2} + 7 M (3 R^{2}) = \frac{181}{2} M R^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365931 Three identical thin rods, each of mass $m$ and length $ℓ$ , are joined to form an equilateral triangular frame. The moment of inertia of the frame about the axis parallel to its one side and passing through the opposite vertex is

1

\frac{3}{2} m ℓ^{2}

2

\frac{5}{4} m ℓ^{2}

3

\frac{5}{2} m ℓ^{2}

4

\frac{5}{3} m ℓ^{2}

Explanation:

Moment of inertia of each of $rod A C$ and $B C$ about the given axis $OO$ ' is
$I_{A C} = I_{B C} = \frac{m ℓ^{2}}{3} \sin^{2} 60^{\circ} = \frac{m ℓ^{2}}{4}$
supporting img
and $M . I$ of rod $A B$ about the given axis $O O$ is
$I_{A B} = m {(\frac{ℓ \sqrt{3}}{2})}^{2} = \frac{3}{4} m ℓ^{2}$
Hence,
$I = I_{A C} + I_{B C} + I_{A B} = \frac{m ℓ^{2}}{4} + \frac{m ℓ^{2}}{4} + \frac{3}{4} m ℓ^{2} = \frac{5}{4} m ℓ^{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365932 Two masses each of mass $M$ are attached to the end of a rigid massless rod of length $L$ . The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is

1

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

2

2 M L^{2}

3

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

4

M L^{2}

Explanation:

The moment of inertia of the system about the given axis is
$I = M {(\frac{L}{2})}^{2} + M {(\frac{L}{2})}^{2} = \frac{M L^{2}}{4} + \frac{M L^{2}}{4} = \frac{M L^{2}}{2}$

PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

365933 Four thin identical rods are joined to form a square as shown. If it is rotated along an axis passing through one side, then its moment of inertia is equal to
supporting img

1

\frac{8}{3} m l^{2}

2

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

3

\frac{6}{3} m l^{2}

4

\frac{5}{3} m l^{2}

Explanation:

$I = \frac{m l^{2}}{3} + \frac{m l^{2}}{3} + m l^{2}$
$I = \frac{5}{3} m l^{2}$

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