Rigid Bodies: Translational Motion and Rotational Motion

Bookmark added to your notes.
View Notes

Rigid Bodies Translational Motion and Rotational Motion in Detail

A rigid body is the one that does not change its shape or deform. In scientific terms, a rigid body is a collection of particles where the distance between its constituent particles does not change when it is in motion.

Though this is not true since some form of deformation happens in all bodies during the course of the movement. Still, in rigid bodies, this deformation is negligible hence considered as not being there. A rigid body displays a variety of motions, and in this article, we will go through the translational and rotational motion of a rigid body.

What is Translatory Motion

In its movement through space, one of the motions that a rigid body undergoes is translatory motion. So, what is meant by translatory motion? The translational motion meaning can be explained with a diagram, as shown below:

(image will be uploaded soon)

The motion through which a body shifts from one point in space to another is translational. As an example, a bullet fired from a gun undergoes a translational movement. When the rigid body moves in a translational motion, the line segment between any two particles of the body remains parallel. A translational motion can be further broken down into two types:

  • Rectilinear Translational Motion - A body moving in a straight line displays rectilinear translational movement. At any time t, the object which is undergoing rectilinear translation occupies a position on the line depicted in the figure below:

(image will be uploaded soon)

If you know the object's position at a given time, you can know the motion of the particle, and it is given by relating the distance x with time t. This motion formula gets more complicated in a two or three dimensional plane like a ball rolling off a table.

  • Curvilinear Translational Motion - Curvilinear translational movement is characterized by a rigid body's movement on a curved surface.

Position, Velocity, and Acceleration of a Rigid body in Translational Motion

  • Position - A rigid body moving through space will have coordinates at any given time. The position of the rigid body is subjective to the observer. Each particle within the rigid body will have its own unique coordinates too.

(image will be uploaded soon)

In the above figure, there are two rigid bodies A and B, which are moving relative to the stationary observer O. Here, SA and SB are absolute paths while SB/A is a relative position. These three positions are associated with each other by the equation:

SB = SA + SB/A.

  • Velocity - A rigid body moving in space will change positions within a specified time. The velocity of the rigid body is given by a change in the position concerning the change in time. So velocity is given by the following derivative:

V = ds/dt; where s is distance and t is time.

In a translational motion, all the particles within the body will also have the same velocity.

  • Acceleration - It is possible that the velocities of rigid bodies moving in space change. This results in acceleration and to find it we make use of the derivative of the rigid body’s velocity or its double derivative to find its position in respect to time;

Acceleration a =dv/dt = d2s/dt2

Rotational Motion of a Rigid Body

Rotary motions happen only in rigid bodies. Few examples of rotational movement are the motions of the earth or the motion of planets around the sun.

A pure rotational motion is when a body spins around a fixed internal axis. In a rotational motion, all the constituent particles of the rigid body undergo circular motion about the common axis.

(image will be uploaded soon)

Common examples of Rotational Motion are:

  • The motion of motors, gears, wheels, top, ferris wheel, etc.

  • The motion of a helicopter’s blades.

  • Move a door that is swiveling on its hinges when you open and close it.

Rotational Motion of Earth

Earth is continuously spinning about its axis. The Earth’s axis is an imaginary line running from its North pole to the South pole. Earth’s rotation is its spinning motion along this imaginary axis. Earth, along with its rotational motion, also orbits or revolves around the Sun.

(image will be uploaded soon)

Earth takes precisely 23 hours, 56 minutes, and 4 seconds to make one complete rotation about its axis. But as the Earth is also revolving around the Sun, it needs to turn a little bit more to come at the same place relative to the Sun. Hence the length of a day is 24 hours. The Earth’s rotational speed at the equator is approximately 1700 kilometers per hour. This speed goes down as we move away from the equator, and at the poles, it is almost nothing.

Angular Displacement

The distance moved by the particles of a rotating rigid body gives its angular displacement. It is measured in radians, and all the particles go through this except for the ones which are present on the fixed axis of rotation. Particles on the fixed axis do not undergo any angular displacement.

(image will be uploaded soon)

The further a particle is from the fixed axis, the more is its angular displacement. The equation to find this displacement is:

Distance = θ * r, Here r is the distance of the particle from the axis of rotation.

FAQ (Frequently Asked Questions)

Q1. What is Rotational Inertia?

Ans. Any object which can be rotated possesses this property called rotational inertia or moment of inertia. This is a scalar value (not a vector) which denotes the difficulty of changing the rotational velocity of an object that is rotating around an axis. Rotational inertia in rotational physics is analogous to mass in linear motion. Just like we gauge an object’s mass (as per Newton’s 2nd law) to know its resistance to change, inertia plays the same role in rotating bodies. The rotational inertia is also referred to as the second moment of mass. The rotational inertia is dependent on the radius of rotation and is given by:

Rotational inertia I = m* r2, where m is the mass of the rotating body and r is the radius of rotation. The moment of inertia is measured in Kg.m2.

Q2. What is the Difference Between Rotation and Revolution?

Ans. An object's movement on its internal axis is called rotational motion while the motion of an object in a circular path along an external axis is called revolution. The most common example is the movement of planet Earth on its axis is rotation (this causes the phenomena of day and night). Earth's movement around the Sun is a revolution (this causes change weather like summers and winters).