JEE Main 2025-26 Rotational Motion Mock Test: Practice & Solutions

The four main rotational kinematic equations (analogous to linear motion equations) under constant angular acceleration are:
1. θ = ω₀t + ½αt²
2. ω = ω₀ + αt
3. ω² = ω₀² + 2αθ
4. θ = ((ω + ω₀)/2)t
Where θ is the angular displacement, ω is the final angular velocity, ω₀ is the initial angular velocity, α is the angular acceleration, and t is time.

Moment of inertia measures an object's resistance to change in its rotational motion, similar to mass in linear motion. It depends on the distribution of mass around the axis of rotation. For example:
Solid sphere: (2/5)MR²
Ring: MR²
Rod (about center): (1/12)ML²
Here, M is mass, R is radius, and L is length. For composite bodies, add the individual moments using the parallel axis theorem if needed.

Angular momentum (L) is defined as the product of moment of inertia (I) and angular velocity (ω): L = Iω. The law of conservation of angular momentum states that if no external torque acts on a body, its angular momentum remains constant. This principle explains phenomena like a figure skater spinning faster when pulling in arms during a spin.

Mechanical equilibrium for rigid bodies requires:
1. Net force on the body is zero (translational equilibrium): ΣF = 0
2. Net torque about any axis is zero (rotational equilibrium): Στ = 0
Both conditions must be satisfied simultaneously for a rigid body to be in complete equilibrium.

Torque (τ) is the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis. Torque is calculated as:
τ = r × F
where r is the perpendicular distance from the axis of rotation to the line of action of the force, and F is the force applied. In scalar form, τ = rFsinθ, where θ is the angle between the force and position vectors.

Rolling motion occurs when a body like a wheel moves such that it undergoes both translational and rotational motion simultaneously. The point in contact with the ground has zero velocity relative to the surface. The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies: K.E. = (1/2)Mv² + (1/2)Iω²

Centripetal force is the real force that acts towards the centre of the circle, keeping an object in circular motion. Centrifugal force is an apparent or pseudo-force experienced in a rotating frame that seems to act outward away from the center. Only centripetal force actually acts in an inertial reference frame.

To solve rotational motion numericals for competitive exams:
• Identify known and unknown values (moment of inertia, angular velocity, radius, mass, etc.)
• Use appropriate formulas: for torque, kinematics, energy, angular momentum, etc.
• Apply the parallel axis theorem or perpendicular axis theorem if the axis is not the centre of mass
• Draw free body diagrams to resolve forces and torques
• Ensure units are consistent and answer matches the required format

Common examples of rotational motion include:
• Rotation of the wheels of a car or bicycle
• Spinning of a ceiling fan
• Movement of planets around the sun
• Rotation of blades in a mixer or washing machine
These examples illustrate the application of angular velocity, torque, and moment of inertia in practical scenarios.

Radius of gyration (k) is a measure of how mass is distributed with respect to the axis of rotation. It is defined as I = Mk², where I is the moment of inertia and M is the total mass. The radius of gyration simplifies calculations and helps in comparing different shapes and mass distributions.

Rotational motion is frequently asked in entrance exams because it combines concepts from Newtonian mechanics, develops problem-solving skills, and reinforces understanding of real-world applications such as machines, vehicles, and planetary motion. Mastery of rotational dynamics is essential for scoring well in JEE, NEET, and MHT CET physics sections.