equilibrium and center of gravity

Learning objectives

• Understand the meaning of center of gravity
• Understand the elements of equilibrium and center of gravity

Center of gravity

The center of gravity of a body refers to the point of application of the resultant force as a result of the attraction of the earth. It is this point where all the weight of a body appears to act on. The force that results is called the weight of the body. To find the weight of an object, multiply mass by gravity.

Knowing the center of gravity of an object is important since it predicts the behavior of a body in motion when the force of gravity acts upon it. The center of gravity is also important in the design of static structures like bridges and buildings.

In a gravitational field that is uniform, the center of gravity is similar to the center of mass. However, you should note that these two points do not always coincide. For example, the center of mass of the moon is very close to the moon’s geometric center. However, the moon’s center of gravity is slightly away from the center of the moon towards the earth, as a result of stronger gravitational force on the near side of the moon.

If an object is symmetrical in shape and is made of homogenous material, the center of gravity coincides with the geometric center of the object. However, for an object that is asymmetrical and is made up of different materials that have different masses, the center of mass will be away from the geometric center of the object. In irregularly shaped or hollow bodies, the center of gravity is located at a point, external of the object.

Difference between center of gravity and center of mass

It is common for many people to assume that the center of gravity and the center of mass are the same. However, the truth is that they are different.

The center of mass refers to a point where the distribution of mass is equal in all directions. The center of mass is not dependent on the field of gravity. The center of gravity on the other hand is the point where the weight of an object is equal in all directions, and it is dependent on the field of gravity.

However, the center of mass and the center of gravity of an object can lie at the same point if the gravitational field is uniform.

Who made the discovery of center of gravity?

The center of gravity was discovered by Archimedes of Syracuse.

What effect does the center of gravity have on balance?

The center of gravity determines the stability of objects. Objects that have lower center of gravity are more stable than objects that have a higher center of gravity. Objects that have very high center of gravity topple over when pushed. Racing cars have low centers of gravity to enable them to negotiate corners without turning over.

What about the center of gravity in our body?

In an anatomical position of our bodies, the center of gravity is located anterior to the 2^nd sacral vertebra. However, note that, since humans do not remain in an anatomical fixed position, the exact location of the center of gravity changes with the position of the limbs, and the body.

Center of gravity of regular shapes

A uniform body (a body that has its weight distributed uniformly) has its center of gravity located at the geometrical center of the body. For example, a meter rule that is uniform has its center of gravity at the 50cm mark.

The center of gravity of regular shapes may also be determined by the construction. For example;

• For rectangular and square objects, diagonal lines are constructed. The center of gravity is the point that these two diagonals intersect.
• For triangular bodies, we construct perpendicular bisectors of the sides. The center of gravity is the point that the three lines intersect.
• For circular bodies, diameters are constructed. The center of gravity is the point that these lines intersect.

Example

A uniform meter rule is balanced at a mark of 20 cm when a 1.2N load is hung at the zero mark. Calculate the weight and mass of the meter rule.

Solution

Start by drawing a diagram that shows all the forces that act on it.

At balance (equilibrium), the sum of clockwise moment = sum of anticlockwise moment

While calculating clockwise moments, you should note that the meter rule has its weight acting at the 50cm mark. Clockwise moments equals the weight of the meter rule multiplied by the distance between the center of the meter rule and the point of the pivot. Therefore, clockwise moments equals weight * 0.3 meters. Anticlockwise moments equals weight of the load multiplied by the distance between the load and the pivot. Therefore, anticlockwise moments equals 1.2 Newtons * 0.2 meters.

W * 0.3m = 1.2N * 0.2m

0.3W = 0.24

W = 0.24/0.3 = 0.8N

Therefore, the weight of the meter rule is 0.8 Newtons.

Determine the reaction on the pivot:

Total upward force = total downward force

R = 1.2 + W

R= 1.2 + 0.8

R= 2 Newton

States of equilibrium

EQUILIBRIUM STATE. This refers to the state of balance of a body. These states are of three different types;

• Stable equilibrium. A body is said to be in this state if it returns to its original position after a slight displacement. The funnel does not topple over because the line of action of weight falls in the base of the funnel. This is applied in modern buses where the luggage compartments are placed in the lower part to lower the position of the center of gravity. This helps increase the stability of the bus.
• Unstable equilibrium. A body is said to be in this state if, after a slight displacement, it does not return to its original positions but occupies a new position. Buses that carry heavy luggage on the top carrier raise the position of the center of gravity. This makes the bus to be unstable.
• Neutral equilibrium. A body is said to be in this state if it occupies a new position similar to the original position when slightly displaced.

Conditions for equilibrium

• The sum of the clockwise moments about any point equals the sum of anticlockwise moments about the same point.
• The sum of forces on the body in one direction equals the sum of forces acting on the body in the opposite direction.

Factors affecting the stability of a body

Two factors affect the stability of a body. They are;

• The base area of a body. Bodies with wider bases are more stable. Narrow bases have less stability.
• The position of the center of gravity. Bodies are more stable when the center of gravity is lower.

Application of stability

• Containers that hold liquids like the laboratory’s conical flask with a broad base to improve stability.
• Automotive applications. For example,racing cars have wider wheels and lower positions of center of gravity than ordinary cars.
• Most buses carry their cargo below the passenger level instead of at the roof rack to lower the center of gravity.
• Aeronautics. Aircrafts are designed such a way that the center of gravity falls within a given limit to make them stable. This makes it easy for them to maneuver
• Body motion. Body movement is also dependent on center of gravity. By understanding about the center of gravity and stability, we understand more about human locomotion.