Test: Moment of Inertia and Centroid - Mechanical Engineering MCQ

Concept:
Moment of inertia of circular plate,


Moment of inertia of Square plate,


Calculation:
The ratio of the moment of inertia of a circular plate to that of a square plate is, Which is less than 1.
Important Points
The following table shows the Second moment of inertia of different shapes

Moment of inertia of an area or Second moment of area (MI):

• MI of a body about any axis is defined as the summation of the second moment of all elementary areas about the axis.
• I = Σ(A × d²)

Unit: m⁴ or mm⁴ or cm⁴
For a square of side a or b:

Moment of inertia is 

Polar moment of inertia of Solid cylinder:

Polar moment of inertia of Hollow Cylinder:

where outer diameter = D and inner diameter = d

The moment of inertia (I1) of a circular plate of diameter “D” is given by:

The moment of inertia (I₂) of the square plate of side equal to the diameter “D” of the circular plate is given by:

• Parallel axis theorem: Moment of inertia of a body about a given axis I is equal to the sum of moment of inertia of the body about an axis parallel to given axis and passing through centre of mass of the body I_o and Ma², where ‘M’ is the mass of the body and ‘a’ is the perpendicular distance between the two axes.

⇒ I = I_o + Ma²
Explanation:

• For a uniform rod with negligible thickness, the moment of inertia about its centre of mass is:

Where M = mass of the rod and L = length of the rod
∴ The moment of inertia about the end of the rod is

The CG of a semicircular plate of  r radius, from its base, is


Calculation:
Given:
r = 33 cm


∴ the C.G. of a semicircular plate of 66 cm diameter, from its base, is 14 cm.
Additional Information
C.G. of the various plain lamina are shown below in the table. Here x̅  & y̅  represent the distance of C.G. from x and y-axis respectively.

Moment of inertia:
Moment of inertia is a measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the element’s distance from the axis.

Moment of inertia of a thin spherical shell of mass M and radius R about its diameter.
I = (2/3)MR²
Additional Information
Moment of inertia of some important shapes:

Moment of inertia:

• The moment of inertia of a rigid body about a fixed axis is defined as the sum of the product of the masses of the particles constituting the body and the square of their respective distances from the axis of the rotation.
• The moment of inertia of a particle  is

⇒ I = mr²
Where r = the perpendicular distance of the particle from the rotational axis.

• Moment of inertia of a body made up of a number of particles (discrete distribution)

Rotational kinetic energy: 

• The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as -

Where I = moment of inertia and ω = angular velocity.

Explanation:

• The moment of inertia of the ring about an axis passing through the center and perpendicular to its plane is given by
  ⇒ I_ring = MR²
• Moment of inertia of the disc about an axis passing through center and perpendicular to its plane is given by -
  
• As we know that mathematically rotational kinetic energy can be written as
  
• According to the question, the angular velocity of a thin disc and a thin ring is the same. Therefore, the kinetic energy depends on the moment of inertia.
• Therefore, a body having more moments of inertia will have more kinetic energy and vice - versa.
• So, from the equation, it is clear that,
  
• The ring has higher kinetic energy.

Important Point

The centroid of the volume is the point where total volume is assumed to be concentrated. It is the geometric centre of a body. If the density is uniform throughout the body, then the center of mass and center of gravity correspond to the centroid of volume. The definition of the centroid of volume is written in terms of ratios of integrals over the volume of the body.

The axis of reference is the axis about which moment of area is taken. Most of the times it is either the standard x or y axis or the centeroidal axis.