Informative line

Moment Of Inertia Of Point Mass System

The axis of rotation is the line on which the centers of circles lie and it is perpendicular to the plane of circles. Calculate the moment of inertia of a 1 kg point mass placed at a point P (2 m, 4 m), taking x-axis as the axis of rotation.

Axis of Rotation of a Rigid Body

  • When a body rotates, all the particles of the body perform circular motion.
  • The axis of rotation is the line on which the centers of circles lie and it is perpendicular to the plane of circles.

Illustration Questions

The axis of rotation for a hinged door will be

A

B

C

D

×

Since, the centers of all the circles lie on the hinged edge of the door. Therefore, the axis of rotation is the vertical line passing through the hinges.

image

The axis of rotation for a hinged door will be

A image
B image
C image
D image

Option B is Correct

Distance from the Axis of Rotation

  • Consider a small particle 'A' of a rotating body.

  • To find the perpendicular distance of 'A' from the axis of rotation, visualize the figure showing the path of particle A such that the axis is seen as a point 'P' i.e., the line of sight is the axis of rotation.

  • Then, the distance between 'A' and 'P' would be the perpendicular distance of particle 'A' from the axis of rotation.

Illustration Questions

Four identical masses A, B, C and D, having mass m each, lie on the vertices of a square of side 'a' cm. If the axis of rotation is in the plane of a square, passing through the center of the square as shown, find the perpendicular distance of A, B, C and D from the axis of rotation.

A rA= a/2, rB= a/2, rC= a/2, rD= a/2

B rA= a, rB= a, rC= a, rD= a

C rA= 2a, rB= 2a, rC= 2a, rD= 2a

D None of these

×

Four identical masses A, B, C and D, having mass m each, lie on the vertices of a square of side 'a' cm.

 

image

Visualize the view of the diagram such that the line of sight becomes the axis of rotation.

image

Perpendicular distance of A from P, rA = a/2

Perpendicular distance of B from P, rB = a/2 

Perpendicular distance of C from P, rC = a/2

Perpendicular distance of D from P, rD = a/2

Four identical masses A, B, C and D, having mass m each, lie on the vertices of a square of side 'a' cm. If the axis of rotation is in the plane of a square, passing through the center of the square as shown, find the perpendicular distance of A, B, C and D from the axis of rotation.

image
A

rA= a/2, rB= a/2, rC= a/2, rD= a/2

.

B

rA= a, rB= a, rC= a, rD= a

C

rA= 2a, rB= 2a, rC= 2a, rD= 2a

D

None of these

Option A is Correct

Moment of Inertia

  • Moment of inertia of discrete particles of varying masses of a body about the given axis is defined as \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation.

Illustration Questions

Calculate the moment of inertia of two point masses placed at A and B about the perpendicular axis passing through the center of mass, as shown.

A \(\dfrac {m\ell^2}{2}\)

B \(\dfrac {m\ell^2}{4}\)

C \(\dfrac {2m\ell^2}{5}\)

D \(\dfrac {2m\ell^2}{3}\)

×

To find the axis of rotation, Centre of Mass (CM) is to be calculated.

image image

Consider the point A as origin:

\(x_{CM}=\dfrac {m_1x_1+m_2x_2}{m_1+m_2}\)

where, \(x_1\) and \(x_2\) are the distances of two point masses (\(m_1\) and \(m_2\)) respectively, from origin

\(x_{CM}=\dfrac {m(0)+2m(\ell)}{m+2m}=\dfrac {2\ell}{3}\)

image

Moment of Inertia,

\(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation

\(\therefore\) \(I=m \left ( \dfrac {2\ell}{3} \right)^2+2m \left ( \dfrac {\ell}{3} \right)^2\)

\(=\dfrac {2m\ell^2}{3}\)

image

Calculate the moment of inertia of two point masses placed at A and B about the perpendicular axis passing through the center of mass, as shown.

image
A

\(\dfrac {m\ell^2}{2}\)

.

B

\(\dfrac {m\ell^2}{4}\)

C

\(\dfrac {2m\ell^2}{5}\)

D

\(\dfrac {2m\ell^2}{3}\)

Option D is Correct

Moment of Inertia of Four Particle System

  • Moment of inertia of discrete particles of varying masses of a body about the given axis is defined as \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation.

Illustration Questions

Four identical masses A, B, C and D having mass m each, lie on the vertices of a square of side 'a'. If the axis of rotation is in the plane of the square, passing through the center of the square as shown, find the moment of inertia of the system.

A \(m\dfrac {a^2}{3}\)

B \(m\dfrac {a^2}{2}\)

C \(ma^2\)

D \(\dfrac {2ma^2}{3}\)

×

Visualize the view of the diagram such that the line of sight becomes the axis of rotation.

image

Perpendicular distance of A from P, rA = a/2

Perpendicular distance of B from P, rB = a/2 

Perpendicular distance of C from P, rC = a/2

Perpendicular distance of D from P, rD = a/2

 

Moment of Inertia,

 \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation

 \(\therefore\,I=m \left ( \dfrac {a}{2} \right)^2+m \left ( \dfrac {a}{2} \right)^2+m \left ( \dfrac {a}{2} \right)^2+m \left ( \dfrac {a}{2} \right)^2\)

\(=ma^2\)

Four identical masses A, B, C and D having mass m each, lie on the vertices of a square of side 'a'. If the axis of rotation is in the plane of the square, passing through the center of the square as shown, find the moment of inertia of the system.

image
A

\(m\dfrac {a^2}{3}\)

.

B

\(m\dfrac {a^2}{2}\)

C

\(ma^2\)

D

\(\dfrac {2ma^2}{3}\)

Option C is Correct

Moment of Inertia of a Point Mass

  • Moment of inertia of a point mass M, which is at a perpendicular distance R from the axis of rotation is defined as :

\(I=MR^2\)

Note : R does not depend upon the point of application of force.

Illustration Questions

Calculate the moment of inertia of a 1 kg point mass placed at a point P (2 m, 4 m), taking x-axis as the axis of rotation.

A 8 kg m2

B 16 kg m2

C 5 kg m2

D 2 kg m2

×

Perpendicular distance of the point mass from x-axis is 4 m.

image

Moment of inertia, \(I\) = MR2

\(I\) = 1 (4)2

= 16 kg m2

image

Calculate the moment of inertia of a 1 kg point mass placed at a point P (2 m, 4 m), taking x-axis as the axis of rotation.

image
A

8 kg m2

.

B

16 kg m2

C

5 kg m2

D

2 kg m2

Option B is Correct

Moment of Inertia of Two Point Masses

  • Moment of inertia of discrete particles of varying masses of a body about the given axis is defined as \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation.

Illustration Questions

Calculate the moment of inertia for a two particle system as shown in the figure about the given axis of rotation.

A \(8m\ell^2\)

B \(5m\ell^2\)

C \(9m\ell^2\)

D \(3m\ell^2\)

×

Mass of particle 1, m1 = m

Mass of particle 2, m2 = 2 m

Distance of particle 1 from axis of rotation, \(r_1=\ell\)

Distance of particle 2 from axis of rotation, \(r_2=2\ell\)

image

Moment of Inertia,

 \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation 

 \(I=m_1r_1^2+m_2r_2^2\)

\(=m(\ell)^2+2m(2\ell)^2\)

\(=9m\ell^2\)

image

Calculate the moment of inertia for a two particle system as shown in the figure about the given axis of rotation.

image
A

\(8m\ell^2\)

.

B

\(5m\ell^2\)

C

\(9m\ell^2\)

D

\(3m\ell^2\)

Option C is Correct

Moment of Inertia of Four Point Masses 

  • Moment of inertia of discrete particles of varying masses of a body about the given axis is defined as \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation.

Illustration Questions

Four particles A, B, C and D, each of mass m, are placed at the vertices of a square frame of side 'a'. Find  the moment of inertia for the system about an axis passing through A, in the plane of the square ABCD at an angle \(\dfrac {\pi}{4}\) with the edge AD.

A \(ma^2\)

B \(2ma^2\)

C \(3ma^2\)

D \(4ma^2\)

×

Four particles A, B, C and D, each of mass m are placed at the vertices of a square frame of side 'a'.

image

Perpendicular distance of A from axis of rotation = rA = 0

Similarly,

 \(r_B=a\,sin\dfrac {\pi}{4}=\dfrac {a}{\sqrt2}\),

\(r_C=a\,\sqrt2\),

\(r_D=\dfrac {a}{\sqrt2}\)

image

Moment of Inertia,

 \(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation.

 \(I=m_Ar_A^2+m_Br_B^2+m_Cr_C^2+m_Dr_D^2\)

\(=0+\dfrac {ma^2}{2}+2ma^2+\dfrac {ma^2}{2}\)

\(=3ma^2\)

image

Four particles A, B, C and D, each of mass m, are placed at the vertices of a square frame of side 'a'. Find  the moment of inertia for the system about an axis passing through A, in the plane of the square ABCD at an angle \(\dfrac {\pi}{4}\) with the edge AD.

image
A

\(ma^2\)

.

B

\(2ma^2\)

C

\(3ma^2\)

D

\(4ma^2\)

Option C is Correct

Illustration Questions

Three particles, each of mass 'm', are situated at the vertices of an equilateral triangle ABC of side L as shown. Calculate the moment of inertia for the given system about the axis passing through A, perpendicular to the plane of ABC.

A \(2mL^2\)

B \(4mL^2\)

C \(8mL^2\)

D \(6mL^2\)

×

Axis of rotation is passing perpendicularly through point A.

Perpendicular distance of point A, rA = 0

Perpendicular distance of point B, rB = L

Perpendicular distance of point C, rC = L

image

Moment of Inertia,

\(I=\sum m_i\;r_i^2\)

where, \(m_i\) is the mass of the ith particle and \(r_i\) is the perpendicular distance of the ith particle from the axis of rotation

 \(I=m_Ar_A^2+m_Br_B^2+m_Cr_C^2\)

\(I=m(0)^2+m(L)^2+m(L)^2\)

\(=2mL^2\)

image

Three particles, each of mass 'm', are situated at the vertices of an equilateral triangle ABC of side L as shown. Calculate the moment of inertia for the given system about the axis passing through A, perpendicular to the plane of ABC.

image
A

\(2mL^2\)

.

B

\(4mL^2\)

C

\(8mL^2\)

D

\(6mL^2\)

Option A is Correct

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