HC Verma Solutions Class 11 Chapter 10 - Rotational Mechanics

The HC Verma solutions for Chapter 10 cover all essential concepts of Rotational Mechanics as per the Class 11 syllabus. Key topics include:

• Moment of inertia and radius of gyration.

• Torque and angular momentum.

• Theorems of perpendicular and parallel axes.

• Equations of rotational motion.

• Conservation of angular momentum.

• Equilibrium of rigid bodies.

While related, they are distinct concepts. Rotational motion (kinematics) describes the movement (like angular velocity and acceleration) without considering its cause. In contrast, rotational dynamics explains the cause of this motion, introducing concepts like torque and moment of inertia. The HC Verma solutions clarify this by first showing how to describe motion with kinematic equations and then applying dynamic principles (like τ = Iα) to find the forces and torques involved.

The Moment of Inertia (I) is the rotational equivalent of mass; it represents an object's resistance to any change in its state of rotational motion. It is crucial because it connects the cause of rotation (torque, τ) to the resulting motion (angular acceleration, α) through the fundamental equation τ = Iα. The HC Verma solutions demonstrate its importance by solving problems where the object's shape and mass distribution directly affect its rotational behaviour.

The HC Verma solutions explain the application of these theorems clearly:

• Parallel Axis Theorem: This is used to find the moment of inertia about any axis that is parallel to an axis passing through the centre of mass. The formula used is I = I_CM + Md², where 'M' is the total mass and 'd' is the perpendicular distance between the two parallel axes.

• Perpendicular Axis Theorem: This applies only to 2D planar objects. It states that the moment of inertia about an axis perpendicular to its plane (I_z) is the sum of the moments of inertia about two mutually perpendicular axes in its plane (I_x + I_y).

The solutions provide step-by-step examples for various geometric shapes.

The principle states that if no net external torque acts on a system, its total angular momentum remains constant (L_initial = L_final). This is a powerful problem-solving tool, especially for scenarios involving collisions or changes in a rotating body's shape (e.g., a spinning dancer pulling their arms in). The HC Verma solutions use this principle to find unknown angular velocities or other variables without needing to analyse the complex internal forces, thus greatly simplifying the solution process.

A common mistake is ensuring only that the net force is zero (translational equilibrium) while forgetting to check if the net torque is also zero. For an object to be in complete mechanical equilibrium, both conditions must be met: the vector sum of all forces must be zero (ΣF = 0), and the sum of all torques about any point must also be zero (Στ = 0). The solutions for HC Verma highlight this by systematically checking both conditions for all equilibrium problems.

The solutions tackle torque problems methodically. The first step is to identify the axis of rotation and the point of force application. Next, they determine the lever arm (the perpendicular distance 'r' from the axis to the line of action of the force 'F'). Finally, they apply the formula τ = rFsinθ, carefully using the correct sign convention (e.g., clockwise torques as negative and anti-clockwise torques as positive) to calculate the net torque on the body.