# Center of Gravity and Moment of Inertia

Center of Gravity and Moment of Inertia are the two most important basic concept for Structural engineers. Their knowledge is a must. It is not possible to do any analysis or design without their knowledge. So, let’s read about Center of Gravity and Moment of Inertia in-depth. Read it completely for the spreadsheet that can ease your work for them.

## Center of Gravity and Moment of Inertia

We can consider any body of mass “m” as the group of minute bodies of mass “dm” like dm_{1}, dm_{2}, dm_{3} ……. and so on. Now, every small component “dm” will have a downward weight “dW_{1}”, “dW_{2}”, “dW_{3}” vector.

The centre of gravity of a body is the point of origin of resultant weight vectors of the system of all the parallel weight vector of the infinitesimal particles of the body.”

## Center of Gravity or Centroid?

The centroid is the geometrical center of the body. Like the center, in a case of a circle. Centroid of a triangle, meeting point of diagonal of the rectangle. Since, weight vectors (“dW_{1}”, “dW_{2}”, “dW_{3}” . . . .. ) can also be stated as (dm_{1}g, dm_{2}g, dm_{3}g,…….) where “g” is gravity vector.

**Uniform Lamina** is the one in which within equal areas have equal weight. So, if a uniform lamina has a symmetrical shape. the centroid of lamina will be geometric center of the lamina.

**Moment of Area about a Point** it means the product of area and its centroidal distance from the point.

Click HERE for the spreadsheet for calculating Centre of Gravity.

## Location of CG of some common shapes.

## Moment of Inertia

Moment of Inertia for body about an axis Say O-O is defined as ∑dM*y_{n}^{2}. Where “dM” are small mass in the body and “y” is the distance of each on of them from the axis O-O. Moment of Inertia is strictly the second moment of mass, just like torque is the first moment of force. However, Structural Engineer mostly use the word Moment of Inertia instead of second moment of area.

So, for the second moment of area. Let’s consider a lamina of area “A” and moment of Inertia (the second moment of area) “I”.

Second moment of area for lamina about an axis Say O-O is defined as ∑dA*y

_{n}^{2}. Where “dA” are small areas in the lamina and “y” is the distance of each on of them from the axis O-O.

**The radius of Gyration** will be sqrt(I/A). The radius of gyration represents the distance from a given axis at which if all the elemental parts are placed will not change the moment of Inertia about that axis.

**Parallel Axes theorem and Perpendicular Axes theorem** are very important theorem which can be used for calculating the moment of any lamina easily. So, They also come in handy while finding the moment about different axis if about another axis is known.

#### The perpendicular Axes Theorem

Let’s say, I_{x} and I_{y }are moments of Inertias of a lamina about X-X and Y-Y axis respectively such that X-X and Y-Y are perpendicular to each other. Then moment about axis Z-Z which is perpendicular to both X-X and Y-Y and passes through their point of intersection will I_{z }such that I_{z }= I_{x }+ I_{y}.

#### The parallel Axes Theorem

It states that the moment of Inertia of lamina about any axis in its plane is given by the sum of Inertia about its centroidal axis and the Ar^{2}. Where “A” is the area of lamina and “r” is the distance between centroidal axis and axis considered.

Note: the radius of gyration, parallel axes theorem, prependicular axe theoram all are valid for Moment of Inertia as well as second moment of area.

It is advised to remember the formula for moment of inertia of some commonly used shapes. It will come handy while finding the moment of Inertia or composite sections.

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