Introduction to Rigid Body Motion

Introduction

In earlier chapters, we primarily focused on the motion of point objects or particles. While considering objects as point masses simplifies analysis, it is often not sufficient to describe the motion of real-world objects, especially those that have size and shape and can rotate or deform.

A rigid body is an idealised object in which the distance between any two constituent particles remains constant, regardless of the forces acting on it. In other words, a rigid body does not deform under the influence of external forces. While no real object is perfectly rigid (all materials deform to some extent under stress), many solid objects like wheels, rods, spheres, etc., can be treated as rigid bodies for the purpose of studying their motion under normal conditions.

The motion of a rigid body is more complex than that of a single particle because it can involve both changes in position and changes in orientation. Understanding the motion of rigid bodies is crucial in various fields of physics and engineering, from designing machines to studying the dynamics of celestial bodies.

What Kind Of Motion Can A Rigid Body Have?

The motion of a rigid body can be described as a combination of two basic types of motion: translational motion and rotational motion.

Pure Translational Motion

A rigid body is said to be in pure translational motion if every particle of the body has the same instantaneous velocity. This means that all particles move together, tracing identical paths. The body as a whole simply shifts its position without changing its orientation.

In pure translation, if you track the motion of any point on the rigid body (e.g., its centre of mass), the entire body's motion is described by the motion of that single point. All points on the body cover the same displacement in the same time interval.

Example: A block sliding on a smooth table without rotating, a train moving on a straight track (if we ignore the rotation of the wheels and any vibrations), a car moving in a straight line if we consider the body of the car without the wheels' rotation.

Diagram showing pure translational motion of a rigid body.

(Image Placeholder: A rigid object (e.g., a rectangle) shown at two different positions, displaced along a straight line. Arrows from corresponding points (like corners) on the initial and final positions show identical, parallel displacements.)

Pure Rotational Motion

A rigid body is in pure rotational motion if it rotates about a fixed axis. In this case, every particle of the body moves in a circle, and the centres of all these circles lie on a fixed line called the axis of rotation. The velocity of each particle is tangential to its circular path, and its magnitude depends on its distance from the axis of rotation (particles farther from the axis move faster). All particles have the same instantaneous angular velocity about the axis of rotation.

The axis of rotation can be fixed in space (e.g., a spinning top rotating about its axis while its base is fixed on the ground) or fixed relative to the body but the body is free to move (e.g., a wheel rotating about its axle, where the axle itself might be moving).

Example: A ceiling fan's blades rotating about its central shaft, a spinning wheel on a fixed axle, the Earth rotating about its axis.

Diagram showing pure rotational motion of a rigid body about a fixed axis.

(Image Placeholder: A rigid object (e.g., a disc or a 3D shape) shown rotating about a central fixed axis. Circles are drawn showing the paths of particles at different distances from the axis. Velocity vectors are shown tangential to these circles.)

Combined Translational and Rotational Motion

The most general motion of a rigid body is a combination of translation and rotation. In this case, the body is simultaneously changing its position in space and changing its orientation.

The motion can be viewed as the translation of some reference point (often the centre of mass) plus rotation about an axis passing through that reference point.

Example: A wheel rolling on the ground. The centre of the wheel is translating forward, while the wheel is also rotating about its axle. A ball thrown with a spin. A car moving on a road (the car body translates, while the wheels rotate). A spinning top whose base is not fixed.

Diagram showing combined translational and rotational motion (e.g., rolling wheel).

(Image Placeholder: A wheel rolling along the ground. An arrow shows the forward velocity of the center. Arrows on the rim show the velocity of points on the rim, which is a combination of the translational velocity of the center and the tangential velocity due to rotation.)

In summary, while a point mass only undergoes translation, a rigid body, due to its extent and rigidity, can undergo translation, rotation, or a combination of both. Analysing these motions requires understanding concepts like torque, angular momentum, and moment of inertia, which are rotational equivalents of force, linear momentum, and mass, respectively.