Rigid body

Some things do not change shape.

If you move a rigid body, it can go in two ways. It can move in a straight line. It can also turn or spin.

When it moves in a line, we call it translation. When it turns, we call it rotation.

All the parts of the object move together. This makes it easy to see where the object is. It is fun to watch things move!

A rigid body is a solid object. It does not change its shape. Even if you push it, the parts stay in the same place. The distance between any two points stays the same.

To know where a rigid body is, we look at two things. First, we look at its linear position. This is its place in a line. We often use the center of mass to find this. The center of mass is the middle point of the object's weight.

Second, we look at its attitude. Attitude is another word for orientation. It tells us which way the object is facing.

When these things change, the object moves. Moving in a line is called translation. Turning or spinning is called rotation.

In science, we also study how fast things move. Linear velocity is how fast the object moves in a line. Angular velocity is how fast the object turns. All parts of a rigid body have the same angular velocity. However, different parts might have different linear velocities if the object is spinning.

A rigid body is a special kind of solid object. In science, we call it rigid because its shape does not change. Even if you apply a deforming force, like a push or a squeeze, the object stays the same. The distance between any two points on the object stays constant over time. This means the object does not bend, stretch, or squish. Scientists use this idea to study how objects move without worrying about them changing shape.

To describe where a rigid body is, we need two pieces of information. First, we look at its linear position. This is the place of the object along a line or path. We usually pick one point to represent the whole object, like its center of mass. The center of mass is the middle point of the object's weight. Second, we look at its attitude, which is another word for orientation. This tells us which way the object is facing in space.

When these two parts change, the object is in motion. Moving from one place to another is called translation. This happens along a straight or curved path. Turning or spinning around is called rotation. You can think of these as two different ways to move. A rigid body can do both at the same time. This combined movement is sometimes called roto-translation.

We can also measure how fast these movements happen. Linear velocity is the speed of the object's position in a line. If an object only moves in a straight line without turning, every part moves at the same speed. However, if the object rotates, different parts move at different linear speeds. For example, points far from the center move faster than points near the center. On the other hand, every part of the rigid body shares the same angular velocity. This is the speed at which the object's orientation changes.

Scientists use different math tools to track these movements. They might use Euler angles or a rotation matrix to describe orientation. These tools help them map how the object turns compared to something else. In the world of tiny particles, molecules are often seen as rigid bodies. They act like collections of point masses, such as electrons and nuclei. Even in space, we use these rules to understand how things like airplanes move.

In the field of classical mechanics, a rigid body is defined as a solid object that does not change shape. When a deforming force or pressure is applied to the object, the deformation is zero or negligible. This means the distance between any two specific points on the body remains constant over time. This property holds true regardless of any external forces or moments acting upon it. Because the shape is fixed, scientists can study the motion of these objects without considering how they might bend or squish. This is a key difference between the mechanics of rigid bodies and the mechanics of materials, which studies deformable objects.

To mathematically describe the position of a rigid body, we must account for all the particles that compose it. Since the body is rigid, we can simplify this by noting that all particles maintain the same relative distance to one another. In theory, knowing the position of just three non-collinear particles is enough to reconstruct the position of the entire body. However, a more convenient approach is to use two distinct components: linear position and angular position. The linear position refers to the location of a specific reference point, often the center of mass or the centroid. The angular position, also known as orientation or attitude, describes how the body is turned in space.

When these two components change over time, the body undergoes motion. If the body moves from one location to another, it is performing translation. This movement can follow a rectilinear path, which is a straight line, or a curvilinear path, which is a curved line. If the body turns or spins, it is performing rotation. When an object performs both translation and rotation simultaneously, the process is called roto-translation. To track these changes, scientists use various numerical methods to define orientation. These include sets of three Euler angles, a quaternion, or a direction cosine matrix, which is also called a rotation matrix.

Velocity describes the rate at which position changes, and it is measured relative to a frame of reference. Linear velocity is a vector quantity representing the time rate of change of the linear position. In purely translational motion, every point on the rigid body moves with the exact same velocity. However, when rotation is involved, the instantaneous velocity of different points will generally vary. Two points will only share the same instantaneous velocity if they lie on an axis parallel to the instantaneous axis of rotation. In contrast, every point on a rigid body experiences the same angular velocity. Angular velocity is a vector quantity describing the angular speed of the change in orientation.

Acceleration is the next step in describing motion, representing the time derivative of velocity. For any point fixed on a rigid body, its acceleration can be calculated based on the body's angular acceleration and its angular velocity. If a point is actually moving within the rigid body itself, its total velocity is a combination of the body's motion and the point's own motion. This complex relationship allows physicists to predict exactly how any part of a moving object will behave. These calculations are essential for understanding everything from simple spinning tops to complex mechanical systems.

While the concept of a rigid body is useful, it has limits in different areas of physics. In the study of special relativity, a perfectly rigid body cannot exist. This is because objects can only be assumed to be rigid if they are not moving near the speed of light. At those extreme speeds, the mass would become infinitely large. In the realm of quantum mechanics, the concept changes slightly. There, a rigid body is often viewed as a collection of point masses. For example, molecules are frequently treated as rigid bodies, specifically as rigid rotors, consisting of point masses like electrons and nuclei.

Understanding rigid body mechanics is vital for many scientific and engineering applications. It allows us to model the flight dynamics of airplanes by defining orientation through orthogonal unit vectors. For instance, one vector might point forward along the wing's chord line, while others define the side and vertical directions. This mathematical framework helps engineers predict how vehicles will react to forces in the air. By treating complex machines as collections of rigid parts, we can master the physics of movement in our world.