Centre of Mass of a Rigid Body

Introduction

In a rigid body, the centre of mass is fixed in reference towards the body, and it will be positioned at the centroid if the body has a uniform density based on its centre of gravity. The centre of mass of open-shaped and hollow objects can occasionally be found outside the physical body. The centre of mass or its system of particles may not match the position of any one member of the system in the event of a distribution of autonomous entities, such as in the case of the planets in the solar system.

When we define the centre of mass, it is indeed a valuable point of reference for mechanics that involve estimating masses scattered in space due to its centre of gravity, such as planetary bodies’ linear and angular momentum and rigid body dynamics. The equations of motion of planets are stated as point masses situated at the centres of mass in celestial mechanics.

What is a Rigid body?

A rigid body (also referred to as a rigid object) is a solid entity having zero or minimal deformation, such that it may be ignored in physics. Regardless of external forces or moments acting on a rigid body, the distance between any two given locations remains constant throughout time. A rigid body is generally perceived as a continuous mass distribution.

A rigid body does not exist under special relativity, and things may only be assumed to be rigid if they are not travelling nearly at the speed of light. A rigid body is commonly thought of, in quantum mechanics, as a collection of point masses. Molecules, for example, are frequently seen as rigid bodies since they are made up of point masses such as electrons and nuclei (see the classification of molecules as rigid rotors).

Centre of Gravity of a Rigid Body

Every rigid body is composed of a number of systems of particles. Gravitational force pulls such point masses towards the Earth’s core. Owing to the Earth being huge in comparison to any real hard body we encounter in everyday life; these forces appear to be operating in lockstep in the downwards direction.

The sum of these opposing forces always operates via a single point. The body’s centre of gravity is located at this location (with respect to Earth). The centre of gravity of a body is the place where all of the body’s weight operates, regardless of its location or orientation. When the gravitational field is homogenous across the body, the rigid body’s centre of gravity and mass coincide.

We may also use trial and error to calculate the centre of gravity of a uniform lamina, even if it has an unusual shape. When the lamina is turned at the centre of gravity, where the net gravitational force acts, the lamina stays horizontal. When a body’s centre of gravity is supported by the torques operating on it, the rigid body’s point masses add up to zero. Furthermore, the pivot’s usual response force compensates for the weight. As the body is in static equilibrium, it stays horizontal.

Kinetics

To represent the linear motion of the body, any point that is firmly attached to the body can be utilised as a reference point (origin of coordinate system L) (Depending on the decision, the linear location, velocity, and acceleration vectors will be different.)

However, depending on the application, the system’s centre of mass would be a better option. This has the simplest motion for a body moving freely in space. It is a place where translational motion is zero or simplified, such as on an axle or hinge, or at the centre of a ball and socket joint, for example.

When the centre of mass is utilised as a reference point, the following happens:

• The circular motion has no effect on the (linear) momentum. It is equal to the entire mass of the rigid body times the translational velocity at any given instant.
• With regard to the centre of mass, the angular momentum is the same as it is without. It is equal to the inertia ten times the angular velocity at all times.
• When the angular velocity is expressed with respect to a coordinate system coinciding with the body’s principal axes, each component of the angular momentum is a product of the torque and the inertia tensor times the angular acceleration; the angular momentum is given by the moment of inertia (a primary value of the inertia tensor) times the appropriate component of the angular velocity.
• In the absence of external forces, possible movements include constant-velocity translation, stable rotation around a fixed primary axis and torque-free precession.
• The net external force on a rigid body is always equal to the total mass times the translational acceleration (i.e., Newton’s second rule pertains to translational motion even though the net external torque is nonzero and/or the body rotates).
• The sum of translational and rotational energy is the total kinetic energy.

Centre of Mass of a Two-particle System

Firstly, choose O as the origin of the coordinate axis.

1. Position of centre of mass m1

m1=m2dm1+m2

1. Position of centre of mass m2

m2=m1dm1+m2

1. The isolated system’s centre of mass has a constant velocity.
2. It indicates that if an isolated system is originally at rest, it will remain at rest, or if it is initially in motion, it will travel at the same velocity.
3. The form, size and distribution of the body’s mass influence the position of the centre of mass.
4. An object’s centre of mass does not have to be located within the object.

Applications

• For the analysis of robotic systems
• Animals, people and humanoid systems require biomechanical study
• For the investigation of extraterrestrial things
• To comprehend the odd movements of hard substances
• For the design and development of gyroscopic sensors and other dynamics-based sensors
• For the design and development of a variety of automotive stability-enhancing applications
• To improve the visuals of video games using stiff bodies

Conclusion

If each particle in a rigid body experiences the same displacement in the same direction over a certain time interval, the body executes a pure translational motion.

If each particle of a rigid body travels in a circle and the centres of the circles lie on a straight line termed the axis of rotation, the rigid body would perform a pure rotational motion.

A rigid body is one in which the relative spacing between the particles of a system does not vary when force is applied. A rigid body’s general motion is made up of translational and rotational movements.