Define center of gravity.

Short Answer:

The center of gravity is the point in a body where the entire weight of the body can be considered to act, irrespective of the body’s position. It is the average location of the gravitational force on all the particles of a body. For a uniform body with symmetrical shape, the center of gravity lies at its geometric center. However, for irregular or non-uniform bodies, it may not coincide with the centroid. The position of the center of gravity depends on the shape, mass distribution, and density of the material.

Detailed Explanation :

Center of Gravity

In Engineering Mechanics, the center of gravity (C.G.) is one of the most important concepts used to understand the stability and balance of bodies. Every object, regardless of its shape or size, has a point where its total weight may be assumed to be concentrated — this point is called the center of gravity.

When a body is supported or suspended at its center of gravity, it remains in perfect balance and does not tend to rotate in any direction. In other words, the center of gravity represents the point where the resultant of all gravitational forces acting on the particles of the body passes through.

This concept is widely used in mechanical, civil, and structural engineering to ensure stability in machines, vehicles, and structures such as bridges and towers.

Definition

The center of gravity is defined as:
“The point through which the entire weight of a body may be considered to act, irrespective of the position of the body.”

Alternatively,
“It is the point at which the resultant weight of the distributed mass of a body acts under the influence of gravity.”

Mathematically, for a system of particles, the coordinates of the center of gravity (x̄, ȳ, z̄) are given by:

where,

  • = Weight of the individual particle,
  • = Coordinates of each particle.

For a uniform solid body, the weight is distributed evenly, and hence the center of gravity coincides with the centroid.

Physical Meaning

The center of gravity represents the point of action of the resultant weight of a body. When a body is supported at its C.G., it remains in stable equilibrium because the gravitational pull acts vertically downward through this point.

For example:

  • A uniform rectangular plate has its center of gravity at the intersection of its diagonals.
  • A triangular plate has its C.G. at the point where the medians intersect (two-thirds from the vertex).
  • In irregular shapes, the C.G. may lie outside the material boundary, such as in a ring or hollow sphere.

Center of Gravity of Common Bodies

  1. Rectangle or Parallelogram:
    The C.G. lies at the intersection of diagonals.
  2. Triangle:
    The C.G. lies at the point where the medians intersect (distance = h/3 from the base).
  3. Circle or Sphere:
    The C.G. lies at the geometric center.
  4. Cone:
    The C.G. lies on the axis at a distance of h/4 from the base.
  5. Semicircular Area:
    The C.G. lies on the axis of symmetry at a distance of 4r/3π from the base.

Factors Affecting the Position of Center of Gravity

The location of the center of gravity depends on several factors:

  1. Shape of the Body:
    Regular shapes (like circles or rectangles) have their C.G. at the center, while irregular shapes have it at asymmetrical points.
  2. Mass Distribution:
    In non-uniform bodies, where mass is not evenly distributed, the C.G. shifts toward the heavier portion.
  3. Density of Material:
    If the material has varying density, the C.G. will not coincide with the geometric center.
  4. External Modifications:
    Cutting, adding, or removing parts of a body will change its mass distribution and hence its center of gravity.

Determination of Center of Gravity

The center of gravity of a body can be determined in different ways depending on its shape and structure.

  1. Analytical Method:
    For regular and composite bodies, the coordinates of the center of gravity can be calculated mathematically using the formula:
  1. Experimental Method:
    For irregular plates, the C.G. can be located by:
  • Suspending the plate freely from different points.
  • Drawing vertical lines along the string in each suspended position.
  • The intersection of these lines gives the center of gravity.

Relation Between Center of Gravity and Centroid

The centroid and the center of gravity are closely related but not identical.

  • The centroid refers to the geometric center of an area or volume, independent of weight.
  • The center of gravity refers to the point through which the resultant weight of the body acts, considering gravitational forces.

For a body of uniform density, the centroid and center of gravity coincide because weight distribution is uniform.
However, for non-uniform bodies (like varying thickness or density), they are different.

Example:
A steel plate with varying thickness will have its center of gravity shifted toward the thicker region, even though the centroid remains the same.

Importance of Center of Gravity in Engineering

  1. Stability of Structures:
    • The stability of buildings, towers, and vehicles depends on the height and location of the C.G.
    • A lower C.G. increases stability, while a higher C.G. makes a body easier to topple.
  2. Design of Vehicles:
    • In cars, buses, and trains, the C.G. is kept low to prevent overturning on sharp turns.
  3. Aircraft and Ships:
    • Proper balance and weight distribution are maintained using the center of gravity for safe operation.
  4. Mechanical Systems:
    • In rotating machines, locating the C.G. helps in balancing rotating parts to reduce vibrations.
  5. Construction Engineering:
    • Used in determining the resultant forces and moments acting on beams, columns, and supports.

Practical Example

Consider a solid cone of height .
The C.G. lies on its axis, at a distance of from the base.

Hence, the center of gravity is located 3 cm above the base of the cone.

This principle helps engineers design and balance conical tanks, turbines, and machine parts.

Graphical Understanding

If the shape is irregular, the C.G. can be determined by the plumb line method:

  1. Suspend the object freely and mark a vertical line from the suspension point.
  2. Repeat by suspending from a different point.
  3. The intersection of the two lines gives the center of gravity.

This experimental method is often used in laboratories to find the C.G. of flat plates or cardboard cutouts.

Conclusion

In conclusion, the center of gravity is the point where the total weight of a body acts, making it a critical factor in determining balance, motion, and stability. For uniform bodies, the center of gravity coincides with the centroid, but for non-uniform bodies, it depends on the distribution of mass and density. In engineering applications, maintaining a proper center of gravity ensures stability, safety, and performance of structures and machines. Understanding and locating the center of gravity is therefore essential for all mechanical and structural design work.