Q2. What is the Parallel Force Theory?

The equations used to calculate the centroid of an arbitrary shape are based on the parallel force theory used to calculate the center of gravity (C.G.) of objects. Consider, for example, the system of particles shown below. Each particle has a finite weight and is located at some position in space. This distributed system can be represented by a single large particle that is equal in weight to that of all the particles combined, and is placed at the center of gravity of the system.

Notice that the system of particles can be viewed as a non-concurrent parallel force system. Based on the Cartesian coordinate system shown, the center of gravity is determined by balancing the total moment created by the weight of each individual particle about an axis to that of the resultant weight. For example, considering the moment about the x axis, we have

Which can be rewritten as

If each particle is viewed as a circle of finite area, then the centroid of the distributed system can be calculated by replacing the weight by area in the above equation.

A similar procedure can be used to derive the equations for the other two coordinates of the centroid, i.e., and .

We can also use the theory of parallel forces to derive the equations for centroids of lines and volumes in addition to areas of arbitrary shape.

Whenever a shape cannot be divided into multiple sections of simple geometric shapes. For example, to determine the centroid of the area confined between two intersecting red and blue curves, as shown in Fig. a, we must use the integral form of centroid equations whereas for the area shown in Fig. b we can simply use the simple summation formulation.

Fig. a		Fig. b

Q4. Can two shapes with equal areas have different area moments of inertia?

Yes! The area moment of inertia of a shape depends on its geometric boundary as well as its size.

Q5. Under what condition is the product of inertia of a shape zero?

The product of inertia is defined with respect to two perpendicular axes. If these axes are the principal axes of the shape, then the product of inertia with respect to these axes would be zero.