Introduction
Rotational motion is a fundamental concept in physics that deals with objects rotating about an axis. Unlike linear motion, where objects move along a straight path, rotational motion involves the angular displacement of objects. This topic is especially important for JEE Advanced Physics as it encompasses various principles and laws that are critical for solving complex problems.
Angular Variables
Angular Displacement ($\theta$)
- Definition: The angle through which a point or line has been rotated in a specified sense about a specified axis.
- Unit: Radians (rad)
Angular Velocity ($\omega$)
- Definition: The rate of change of angular displacement.
- Formula: $$\omega = \frac{d\theta}{dt}$$
- Unit: Radians per second (rad/s)
Angular Acceleration ($\alpha$)
- Definition: The rate of change of angular velocity.
- Formula: $$\alpha = \frac{d\omega}{dt}$$
- Unit: Radians per second squared (rad/s²)
Angular quantities are analogous to linear quantities but are specific to rotational motion.
Kinematic Equations for Rotational Motion
Similar to linear motion, rotational motion has its own set of kinematic equations:
- $$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$$
- $$\omega = \omega_0 + \alpha t$$
- $$\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$$
These equations are useful for solving problems where angular acceleration is constant.
Moment of Inertia (I)
Definition
- Definition: The rotational analog of mass for linear motion. It is a measure of an object's resistance to changes in its rotation rate.
- Formula: $$I = \sum m_i r_i^2$$ where $m_i$ is the mass of the $i$-th particle and $r_i$ is the distance from the axis of rotation.
Parallel Axis Theorem
- Formula: $$I = I_{cm} + Md^2$$ where $I_{cm}$ is the moment of inertia about the center of mass, $M$ is the mass of the object, and $d$ is the distance between the center of mass and the new axis.
Perpendicular Axis Theorem
- Formula: $$I_z = I_x + I_y$$ where $I_z$ is the moment of inertia about an axis perpendicular to the plane, and $I_x$ and $I_y$ are the moments of inertia about two perpendicular axes in the plane.
Students often confuse the parallel axis theorem with the perpendicular axis theorem. Make sure to understand the context in which each theorem is applied.
Torque ($\tau$)
Definition
- Definition: A measure of the force that can cause an object to rotate about an axis.
- Formula: $$\tau = r \times F = r F \sin \theta$$ where $r$ is the lever arm, $F$ is the force, and $\theta$ is the angle between $r$ and $F$.
Relation to Angular Acceleration
- Formula: $$\tau = I \alpha$$
Example Calculation: A force of 10 N is applied at the end of a 0.5 m long wrench at an angle of 90°. Calculate the torque.
$$\tau = r \times F = 0.5 , \text{m} \times 10 , \text{N} = 5 , \text{Nm}$$
Rotational Kinetic Energy
- Formula: $$K.E. = \frac{1}{2} I \omega^2$$
Example Calculation: Calculate the rotational kinetic energy of a disc with a moment of inertia of 0.1 kg·m² rotating at 10 rad/s.
$$K.E. = \frac{1}{2} \times 0.1 , \text{kg·m}^2 \times (10 , \text{rad/s})^2 = 5 , \text{J}$$
Angular Momentum ($L$)
Definition
- Definition: The rotational analog of linear momentum.
- Formula: $$L = I \omega$$
Conservation of Angular Momentum
- Principle: If no external torque acts on a system, its total angular momentum remains constant.
- Formula: $$L_{initial} = L_{final}$$
Conservation of angular momentum is crucial for solving problems involving isolated systems.
Rolling Motion
Pure Rolling
- Condition: When an object rolls without slipping, the point of contact with the surface is momentarily at rest.
- Relation: $$v = R \omega$$ where $v$ is the linear velocity and $R$ is the radius of the rolling object.
In pure rolling, the linear velocity of the center of mass is equal to the product of the angular velocity and the radius.
Kinetic Energy in Rolling Motion
- Formula: $$K.E. = \frac{1}{2} I \omega^2 + \frac{1}{2} m v^2$$ where $m$ is the mass of the object.
Example Calculation: Calculate the total kinetic energy of a rolling cylinder with mass 2 kg, radius 0.1 m, and angular velocity 5 rad/s. The moment of inertia of the cylinder is $I = \frac{1}{2} m R^2$.
$$I = \frac{1}{2} \times 2 , \text{kg} \times (0.1 , \text{m})^2 = 0.01 , \text{kg·m}^2$$ $$K.E. = \frac{1}{2} \times 0.01 , \text{kg·m}^2 \times (5 , \text{rad/s})^2 + \frac{1}{2} \times 2 , \text{kg} \times (0.5 , \text{m/s})^2 = 0.125 , \text{J} + 0.25 , \text{J} = 0.375 , \text{J}$$
Summary
Rotational motion is a complex yet fundamental topic in physics, especially for JEE Advanced. Understanding the various angular quantities, kinematic equations, and principles such as torque, moment of inertia, and angular momentum is crucial for solving problems effectively. Always remember to apply the correct theorems and principles based on the context of the problem.
TipPractice solving a variety of problems to become proficient in applying these concepts. Use diagrams to visualize the problem whenever possible.
