Tuesday , October 24 2017

How to Calculate Center of Gravity

In physics, center of gravity is the point where the entire weight of a body can be assumed to be concentrated. There is another term used in science, called center of mass, which is the point where the entire mass of the body can be assumed to be concentrated. We need to calculate the mean position of the mass of the object to find its center of mass. These two points are same only in the case of uniformly distributed gravitational force on the object.

Definition of center of gravity

In short definition, the center of gravity is the geometric mean position of the weight of the object. Whereas center of mass is the mean position of total mass of the object.

A simple definition for kids is that, It is such a point on an object from which if it is allowed to suspended through a string, then the body will be stable and do not topple.Using this point of view, a complex object of weight W can be considered as a particle of weight W at its center of gravity. For many scientific calculations, we can describe the motion of any complex object in terms of the motion of its center of gravity; also the rotation of the object can be considered around its center of gravity.

Calculating center of gravity

Calculating the center of gravity of a simple object like ball is very easy, if it is uniform, its center of gravity will be at its center. But for complex shaped objects, generally calculus is required to calculate the center of gravity. If the object has a line of symmetry, the center of gravity will lie on that line.For a system of objects, we can use the following formula,

Center of gravity = (sum of product of masses and their positions)/ sum of masses of the objects

Center of gravity = \frac{\sum{m_i x_i}}{ \sum{m_i}}

x is the position of each mass in the system, and
i is the nos of mass in the system

calculate center of gravity

 In 2D problems of physics the center of gravity of all the particles will lie in the same plane, making the calculation easier, but in 3D all the three coordinates are involved and it also represents a more practical situation.

Let us assume a system of particles of mass m_1, m_2, m_3 and position of their center of gravity as x_1, x_2, x_3  respectively.

Then the center of gravity of the system can be found by following method.

Center of gravity = \frac{(m_1 . x_1 +m_2 . x_2 + m_3 . x_3)}{(m_1+m_2+m_3)}

Examples of center of gravity

Center of gravity has a huge relevance in our everyday life.

Taller things topple easily than shorter ones, because the center of gravity of a tall object is farther from the surface of earth than the shorter. Thus, racing cars are designed with their center of gravity closer to the ground, since it reduces their chances to tip over while they are at a fast speed.

Also, airplanes and rockets rotate around their center of gravity.

Therefore, center of gravity plays a vital role in various aspects of the real life.

Let us see a calculation example below

Consider a lamppost of height 20 m, equally distributed mass 10 kg. A bird of mass 2 kg is sitting at the top of the lamppost. What would be the center of gravity of the system consisting of the lamppost and the bird?

Since the lamppost is vertically standing and the bird is sitting at the top of it. The axis of the center of gravity of the bird and the post will be same. We can consider the mass of the bird to be concentrated at the top of the lamppost, that is, at 20 m from the ground. The center of gravity of the lamppost will be at its geometric center, that is, 10 m.

Now using,

Center of gravity = \frac{\sum{m_i x_i}}{ \sum{m_i}}

Center of gravity = \frac{(10 x 10 + 2 x 20)}{(10 +2)} = \frac{140}{12} = 11.66 m

Thus the center of gravity of the system will be situated at a height of 11.66 m from ground.

Authored by Kiran Mandia

I am a computer science student at Indian Institute of Technology, Kharagpur; and a writer who loves to read about various topics.I also like making short films.

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