Test: Moment of Inertia - Mechanical Engineering MCQ

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.

The radius of gyration of a body about an axis is a distance such that its square multiplied by the area gives moment of inertia of the area about the given axis. The formula of radius of gyration is given as k² = I/A.

The formula of the moment of inertia is, MOI = ar² where M = mass, a = area, l = length, r = distance.

The theorem of parallel axis states that if the moment of inertia of a plane area about an axis in the plane of area theough the C.G. of the plane area be represented by IG, then the moment of the inertia of the given plane area about a parallel axis AB in the plane of area at a distance h from the C.G. is given by the formula I_AB = Ah² + I_G.

Theorem of perpendicular axis stares that if IXX and IYY be the moment of inertia of a plane section about two mutually perpendicular axis X-X and Y-Y in the plane of the section then the moment of inertia of the section IZZ about the axis Z-Z, perpendicular to the plane and passing through the intersection of X-X and Y-Y is given by the formula I_zz – I_xx = I_yy.

The moment of inertia of a circle, I = πD⁴/64 = 491.07 cm⁴
The area of circle = 78.57 cm,
Radius of gyration = (I/A)^1/2 = 2.5 cm.

The radius of gyration = (length⁴/length²)1/2 = length
So its unit will be m.

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:

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