Rolling motion is a fundamental concept in rotational mechanics, frequently tested in competitive exams like JEE. It combines translational and rotational motion, requiring a solid understanding of both linear and angular quantities, and their interrelationships. This module will guide you through the key principles and problem-solving strategies for rolling motion.

Rolling motion occurs when an object moves along a surface without slipping. This means that at the point of contact between the object and the surface, the instantaneous velocity is zero. This condition is crucial for solving problems.

To tackle rolling motion problems, we typically employ two main approaches: Newton's Laws of Motion (linear and rotational) and the Work-Energy Theorem or Conservation of Energy.

This involves drawing free-body diagrams, applying Newton's second law for linear motion ($\Sigma F = ma_{cm}$) and rotational motion ($\Sigma au = I\alpha$). Remember that the net torque is usually calculated about the center of mass, and the angular acceleration $\alpha$ is related to the linear acceleration $a_{cm}$ by $a_{cm} = R\alpha$ for pure rolling.

The total kinetic energy of a rolling object is the sum of its translational kinetic energy and rotational kinetic energy: $KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2$. Since $v_{cm} = R\omega$, we can express the total kinetic energy in terms of $v_{cm}$ alone: $KE_{total} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I(\frac{v_{cm}}{R})^2 = \frac{1}{2}mv_{cm}^2(1 + \frac{I}{mR^2})$. This form is particularly useful when dealing with energy conservation in problems involving inclines or changes in height.

Problems often involve objects rolling down inclined planes, objects rolling off horizontal surfaces, or scenarios where an object is projected and then starts rolling.

When an object rolls down an incline, static friction acts upwards along the incline. The acceleration depends on the angle of the incline and the object's moment of inertia. Objects with a smaller moment of inertia relative to their mass and radius (like spheres) tend to accelerate faster than objects with larger moments of inertia (like hoops).

If an object rolls off a table, it will follow a projectile path after leaving the edge. The initial velocity for the projectile motion will be the linear velocity of the object at the edge of the table.

Consider a solid sphere rolling down an incline of angle $\theta$. To find its acceleration:

1. Draw a free-body diagram showing gravity, normal force, and static friction.
2. Apply $\Sigma F_x = ma_{cm}$ along the incline: $mg\sin\theta - f_s = ma_{cm}$.
3. Apply $\Sigma \tau_{cm} = I\alpha$ about the center of mass: $f_s R = I\alpha$.
4. Use the no-slip condition: $a_{cm} = R\alpha$, so $\alpha = a_{cm}/R$.
5. Substitute $\alpha$ into the torque equation: $f_s R = I(a_{cm}/R) \implies f_s = Ia_{cm}/R^2$.
6. Substitute $f_s$ into the force equation: $mg\sin\theta - Ia_{cm}/R^2 = ma_{cm}$.
7. Solve for $a_{cm}$: $a_{cm} = \frac{mg\sin\theta}{m + I/R^2} = \frac{g\sin\theta}{1 + I/(mR^2)}$.
8. For a solid sphere, $I = \frac{2}{5}mR^2$, so $a_{cm} = \frac{g\sin\theta}{1 + (2/5)mR^2 / (mR^2)} = \frac{g\sin\theta}{1 + 2/5} = \frac{5}{7}g\sin\theta$.