Dynamics Of Rotational Motion, Rotation About A Fixed Axis, Important Topics For JEE 2025

JEE Advanced Exam Pattern 2024 : Click to Check

Dynamics Of Rotational Motion : A rigid body is defined as an object that has fixed size and shape. In other words, the relative positions of its constituent particles remain constant. In actual, a rigid body does not exist – it is a useful idealization. By the term fixed axis , we mean that the axis must be fixed relative to the body and fixed in direction relative to an inertial position.

A rigid body may have either of the following three types of motions :

(i) Translational motion

(ii) Rotational motion

(iii) Translational plus rotational motion

• In translational motion of the rigid body all particles of the rigid body have same linear displacement, same linear velocity and same linear acceleration. In rest two motions, different particles have different linear displacement, different linear velocity and different linear acceleration.
• As far as translational motion is concerned we do not differentiate between a particle and a rigid body.
• This motion is already discussed in the chapter of kinematics. This is the reason, in the chapter of kinematics, sometimes we write: a particle is moving and sometimes we write: a block (or a body) is moving. Rest two motions are only defined for a rigid body.

JEE Advanced 2024 Chapter Wise Weightage

Dynamics Of Rotational Motion

A rigid body may have either of the following three types of motions :

(i) Translational motion

(ii) Rotational motion

(iii) Translational plus rotational motion

• In translational motion of the rigid body all particles of the rigid body have same linear displacement, same linear velocity and same linear acceleration. In rest two motions, different particles have different linear displacement, different linear velocity and different linear acceleration.
• As far as translational motion is concerned we do not differentiate between a particle and a rigid body.
• This motion is already discussed in the chapter of kinematics. This is the reason, in the chapter of kinematics, sometimes we write: a particle is moving and sometimes we write: a block (or a body) is moving. Rest two motions are only defined for a rigid body. In the present chapter, we shall discuss these two motions of a rigid body.

IIT Allahabad Seat Matrix : Click To Check

Types Of Rotational Motion

Motion of body involving rotation can be classified into the following two categories.

(i) Rotation about a fixed axis or pure rotation.

(ii) Rotation about an axis in translation or combined rotation and translation

IIIT Lucknow Seat Matrix 2024 : Click to Check

Rotation About A Fixed Axis Or Pure Rotation

When a rigid object rotates, points on the object, such as A , B , or C . move in circular paths. An imaginary line perpendicular to plane of circular paths of particles of a rigid body in rotation and containing the centers of all these circular paths is known as axis of rotation. Rotation of ceiling fan, potter's wheel, opening and closing of doors and hands of a wall clock etc. come into this category.

JEE Main Study Material For 2025 : Click to Check

Combined Rotation And Translation

Combined Rotation And Translation : The most general motion of a rigid body is the combination of translational and rotational motions. We have studied pure rotation and pure translation motions but what happens when the two are combined? The most familiar example of combined rotational and translational motion is a rolling wheel. Consider rolling of wheels of a vehicle, moving on straight level road. Relative to a reference frame the axel fixed with the vehicle, the wheel appears rotating about its axel. The rotation of the wheel from this frame is pure rotation about fixed axis. But relative to a reference frame fixed with the ground, the wheel appears rotating about the moving axel, therefore, rolling of a wheel is superposition of two simultaneous but distinct motions-rotation about the fixed axel and translation of the wheel along the level road.

JEE Advanced Cutoff 2023 : Click to Check

Rotation Of A Rigid Body About A Fixed Axis

When a body is rotating about a fixed axis, any point P located in the body travels along a circular path. Before, analyzing the circular motion of point P , we will first study the angular motion properties of a rigid body.

Angular Motion : Since, a point is without dimension, it has no angular motion. Only lines or bodies undergo angular motion. Let us consider the angular motion of a radial line r located with the shaded plane.

Angular Position : ∙ The angular position of r is defined by the angle θ, measured between a fixed reference line OA and r .

Angular Displacement : The change in the angular position, often measured as a differential d θ is called the angular displacement. (Finite angular displacements are not vector quantities, although differential rotations d θ are vectors). This vector has a magnitude d θ and the direction of d θ is along the axis.

• Specifically, the direction of d θ is determined by right hand rule; that is, the fingers of the right hand are curled with the sense of rotation, so that in this case the thumb or d θ points upward.

Most Important Do Or Die Chapters For JEE Main 2025 : Click to Check

Angular Velocity In Rotational Motion

Angular Velocity : The time rate of change in the angular position is called the angular velocity . Thus,

It is expressed here in scalar form, since its direction is always along the axis of rotation, i.e., in the same direction as

Angular Acceleration In Rotational Motion

Angular Acceleration : ∙ The angular acceleration α measures the time rate of change of the angular velocity. Hence, the magnitude of this vector may be written as,

JEE Main Marks Vs Percentile : Click to Check

Instantaneous axis of rotation

• In combined translational and rotational motion, we cannot have a permanent axis of rotation. However, there may be a point inside the given body (or extended body) which can be momentarily at rest. This is what we call instantaneous axis of rotation.
• Consider a rod which remains always in the same vertical plane with its ends in contact with smooth wall and floor as shown in figure. As the rod slides, its centre of mass translates and the rod rotates about a horizontal axis passing through centre of mass. The motion of such an object can be analyzed by making equations for translation and rotation separately, which involves lengthy calculations.

JEE Advanced 2024 Exam : Click to Check

Torque And Angular Acceleration For A Rigid Body

• The angular acceleration of a rigid body is directly proportional to the sum of the torque components along the axis of rotation. The proportionality constant is the inverse of the moment of inertia about that axis, or
• The angular acceleration of a rigid body is directly proportional to the sum of the torque components along the axis of rotation. The proportionality constant is the inverse of the moment of inertia about that axis, or

Thus, for a rigid body we have the rotational analog of Newton’s second law:

…(iii)

Following two points are important regarding the above equations:

(i) The above equation is valid only for rigid bodies. If the body is not rigid like a rotating tank of water, the angular acceleration is different  for different particles.

(ii) The sum in the above equation includes only the torques of the external forces, because all the internal torques add to zero.

JEE Advanced 2024 Exam Date : Click to Check

Torque And Angular Acceleration For A Rigid Body

Table given ahead compares the linear and angular motion with constant acceleration.