What is the Dimensional Formula of Angular Acceleration?

The dimensional formula of angular acceleration is [M⁰L⁰T^-2] or simply [T^-2].

Angular acceleration ($\alpha$) is a fundamental kinematic quantity in rotational motion, representing the rate at which angular velocity changes over time. Understanding its dimensional formula is crucial for analyzing the consistency of physical equations.

Understanding Angular Acceleration

Angular acceleration is defined as the time rate of change of angular velocity ($\omega$).
Mathematically, it is expressed as:
$$ \alpha = \frac{d\omega}{dt} $$
Where:

• $\alpha$ is the angular acceleration.
• $d\omega$ is the change in angular velocity.
• $dt$ is the change in time.

Deriving the Dimensional Formula

To determine the dimensional formula of angular acceleration, we can break it down using the dimensions of its constituent parts:

1. Angle ($\theta$): In physics, an angle is considered a dimensionless quantity. It is the ratio of arc length to radius, both of which have the dimension of length [L]. Therefore, $\theta = \frac{\text{Arc Length}}{\text{Radius}} = \frac{[\text{L}]}{[\text{L}]} = [\text{M}^0\text{L}^0\text{T}^0]$.
2. Angular Velocity ($\omega$): Angular velocity is the rate of change of angular displacement.
   $$ \omega = \frac{d\theta}{dt} $$
   Since angle ($\theta$) is dimensionless ([M⁰L⁰T⁰]) and time ($t$) has the dimension [T], the dimension of angular velocity is:
   $$ [\omega] = \frac{[M^0L^0T^0]}{[T]} = [T^{-1}] $$
3. Angular Acceleration ($\alpha$): As angular acceleration is the rate of change of angular velocity:
   $$ \alpha = \frac{d\omega}{dt} $$
   Substituting the dimension of angular velocity ([T^-1]) and time ([T]):
   $$ [\alpha] = \frac{[T^{-1}]}{[T]} = [T^{-2}] $$
   Including mass [M] and length [L] with a power of zero for completeness, the dimensional formula of angular acceleration is [M⁰L⁰T^-2].

Practical Insights

• Angular acceleration measures how quickly an object's rotation speeds up or slows down. For instance, when a spinning top starts to wobble and slow down, it undergoes angular deceleration (negative angular acceleration).
• It plays a vital role in dynamics, especially in Newton's second law for rotation, where torque ($\tau$) is proportional to angular acceleration ($\tau = I\alpha$, where $I$ is the moment of inertia).
• The SI unit for angular acceleration is radians per second squared (rad/s²).

Dimensional Formulas of Related Physical Quantities

Understanding the dimensions of various physical quantities helps in verifying the correctness of equations and in converting units. Below is a table illustrating the dimensional formulas for several key quantities:

Physical Quantity	Formula	Dimensional Formula
Acceleration (a)	$a = \frac{dv}{dt}$	[L1T-2]
Angular Acceleration (α)	α = dω/dt	[M1T-2]
Momentum (p)	$p = mv$	[M1L1T-1]
Angular Momentum (L)	$L = rp = mvr = m\omega r^2$	[M1L2T-1]

Note: The dimensional formula for angular acceleration is typically [M⁰L⁰T^-2], indicating independence from mass and length. The inclusion of M¹ in some contexts might arise from specific interpretations or conventions, but for a kinematic quantity like angular acceleration, mass is usually not a component of its fundamental dimensions.

For more information on dimensional analysis, you can refer to resources like Khan Academy's explanation of dimensional analysis.