Rigid body

What is a rigid body?

A rigid body is a material object whose particles always remain in the same relative positions. Therefore, it is an object that does not deform, a quality attributed to the strong cohesion forces that hold the particles in place..

In reality, the particles of any body subjected to external forces tend to vibrate or move, so the object always deforms to some extent, but these effects are usually small.

When this is the case, it can be assumed that the body is rigid and have a very good approximation of its behavior, even if it is an idealization.

Rigid body types

Two types of rigid body can be distinguished:

• Those whose particles come in discrete quantities, that is, they can be counted. For example, two metal spheres joined by a thin and light rod can be considered as a single entity. If the rod is stiff enough not to bend, the system is considered a rigid body.
• Those that are continuous, which means that the particles that make them up are indistinguishable. Everyday objects and nature are good examples: stones, furniture and others, as well as the Earth and other rocky celestial bodies.

Rigid body movement and dynamics

Like objects considered as particles, rigid bodies can translate, rotate and have a more general motion, combining translation and rotation.

To study translation, it is not necessary to analyze the movement of each particle separately, but rather the movement of the center of mass is studied, a point where all the mass of the object is considered to be concentrated..

These translational and rotational movements can be:

• Independent, as in the case of planets, which have a rotational movement around their axis (considered fixed) and a translation movement around the Sun, but the speeds of each are not related..
• Roto-translation, if the angular velocity and the translational velocity of the center of mass are related. In this case the axis of rotation is mobile, as in the case of the cylinder that rolls down a steep slope without slipping..

The dynamics of the rigid body

The following quantities are relevant in the dynamics of the rigid body:

Mass center

The center of mass is the point where all the mass of the body is considered to be concentrated. If it is a homogeneous and symmetric body, such as a sphere, the center of mass coincides with the geometric center.

Moment of inertia

This scalar magnitude is the value of the rotational inertia or resistance that an object opposes to rotate around a certain axis. It depends entirely on the geometry of the object and its mass and therefore, in certain cases, it is easier to rotate around certain axes than others..

For bodies with well-defined geometric shapes, there are tables with the moments of inertia with respect to the axes of symmetry, for example the one that passes through the center of mass. With this information and the theorems related to the moment of inertia, the moments about other axes can be easily calculated..

Forces and torque or torque

It takes forces to move a body. If the center of mass of the rigid solid is translated, the equation of motion, according to Newton's second law, is:

F_net = M ∙to_cm

Where:

-The net force is F_net

-M is the mass

-The acceleration of the center of mass is to_cm

However, not all the applied forces succeed in causing the object to rotate. This requires the torque or torque, which tells how effective the rotational action of a force is. It is defined as the vector product between the position vector r with respect to a certain point O and the force F in question. It is denoted by the Greek letter τ (in bold type, as it is also a vector):

τ = r × F

In the SI International System, the unit for torque is N⋅m (newtons per meter).

In many cases, rotational motion about an axis through the center of mass is described by an equation analogous to Newton's second law:

Kinetic energy of a rigid solid

The motion of a rigid body is described by translations of the center of mass and rotations around said point, therefore, its kinetic energy has both contributions.

Let K be the kinetic energy of the body, v_cm the velocity of the center of mass, M the mass of the body, I_cm its moment of inertia about the center of mass and ω the angular velocity. It can be shown that the kinetic energy is:

K = ½ Mv_cm^two + ½ I_cm ω^two

It is observed that the second term on the right is the rotational analog of the term on the left. There the moment of inertia plays the same role as the mass, while the angular velocity has the same role as the linear velocity..

Examples in everyday life

Physical pendulum

The physical pendulum or real pendulum is very easy to build: it consists of a rigid solid like a rod or bar, freely oscillating around a horizontal axis. The axis of rotation does not pass through the center of mass of the object and this in principle can have any shape.

This pendulum differs from the simple pendulum, because in the latter the mass that composes it is considered to be punctual.

Bike wheel

Another example of a well-known rigid body is the bicycle wheel, the axis of which passes through the center of mass, which passes through the center of the wheel. As long as it does not lean to one side or turn, the equations of dynamics described above apply to describe its motion..

A bowling ball

The rigid solid model is very well suited to describe the bowling ball's motion on the lane or when it rolls without sliding down the return ramp.

The yoyo

This popular toy is made with a wooden or plastic cylinder and a string wound in a groove that surrounds it..

The cylinder can be modeled as a rigid body in which the tension in the string provides the torque for the turn, while the weight (applied to the center of mass) and the tension are responsible for the vertical acceleration of the center of mass..

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 
2. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed prentice hall.
3.  Katz, D. 2013. Physics for Scientists and Engineers. Foundations and Connections. Cengage Learning.
4. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1. Pearson.
5. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7th. Ed. Cengage Learning.