Angular Acceleration

Quantities are classified as belonging to two distinct worlds, namely the scalar world and the vector world. Scalar quantities include only the magnitudes while vector quantities encompass most magnitude and direction of motion. Every scalar quantity is mapped to an equivalent vector one. Linear velocity and acceleration of the scalar world are mapped to the angular velocity and angular acceleration of the world of vectors. Acceleration is such an important quantity in kinematics. All three kinematical equations involve acceleration. Newton's second law that talks about the force acting on a body, would have no meaning without this amazing quantity called acceleration. The law says that the force acting on a body is equal to the product of the mass of the body and its acceleration.

Not all things travel straight! Some bodies move in a circular fashion too. Owing to their circular nature of motion, their direction of motion does not remain constant. The bodies that move in a straight direction possess linear velocity while the ones moving in a circular manner possess an angular velocity. The angular velocity changes with time giving rise to the concept of angular acceleration.

Angular acceleration is the rate of change of angular velocity with respect to time. It is a vector quantity as it involves both magnitude and direction. Thus we can say that Angular acceleration = (final angular velocity - initial angular velocity)/time. Thus, it is the deviation in the angular velocity denoted by omega, over time. It is the time derivative of the angular velocity. But what is angular velocity? It is the time derivative of the angular distance, with direction perpendicular to the plane of angular motion. In short, angular velocity is the rate of change of angular distance with respect to time and angular acceleration signifies the rate of change of angular velocity over time.

The SI unit of angular acceleration is radians per second squared and is denoted by the symbol alpha. While discussing angular acceleration, we consider circular motion and this is the reason why the unit radian comes into picture. Radians correspond to the scalar quantity 'degrees' that are used to measure angles. Consider a circle with two radii that make it look like having cut a piece of pie. The angle between the two radii, if measured in radians, is the length of the arc formed by the two radii divided by the length of the radius. The SI unit for angular acceleration is derived from this unit of angular measurement.

Angular acceleration vector does not always point in the direction of the angular velocity vector. Imagine a car speeding along a road. While the car's velocity is directed forward, its angular acceleration vectors point along the direction of the axles of the car wheels. In case, there is a clockwise increase in the angular velocity, the angular acceleration vector points away from the observer, while in case of a counter-clockwise increase in angular velocity, the acceleration vector points towards the observer.

We all know that force acting on an object is the product of its mass and its acceleration. On similar lines, Newton's second law can be adapted to rotational motion by replacing force with torque, mass with moment of inertia and linear acceleration with the angular one. For all constant values of torque, the angular acceleration also remains constant, while the angular acceleration changes with time in case of a non-constant torque.

What's interesting about kinematics is that it tries to measure all the 'forces' in nature by taking into account two basic quantities, the 'mass' characteristic to every concrete body and its zeal to change its velocity that is manifested in terms of its 'acceleration'!