Dimensional Formula of Acceleration and Angular Acceleration

The dimensional formula of Acceleration is 

[ M0  L1  T-2 ]

where [M], [L], and [T] are the fundamental quantities: Mass, Length, and Time.

From the above formula, we can derive the dimensions of acceleration as (0, 1, -2).


Derivation of Dimensional Formula of Acceleration

We can derive this formula using the definition of acceleration.

In physics, we can define acceleration as the rate of change in speed. So, mathematically, acceleration is given as


  Acceleration = Speed / time

We know Speed = distance / time. Substituting the value of Speed in the above equation, we get


  Acceleration = ( distance / time ) / time

  Acceleration = distance / time2

  Acceleration = distance1 * time-2

Dimensionally, we use

  • [L] for distance
  • [T] for time

Now the formula becomes


  Acceleration = [L1] * [T-2]

Since the formula doesn’t include the fundamental quantity Mass, we can use [M0], which is equal to 1.


  Acceleration = [M0] * [L1] * [T-2]

Hence, the dimensional equation of acceleration is a = [ M0 L1 T-2 ].

Derivation of Dimensional Formula of Acceleration

What is the dimensional formula of angular acceleration?

The dimensional formula of angular acceleration is

[M0 L0 T-2]

where [M], [L], and [T] are the fundamental quantities: Mass, Length, and Time.

Derivation

We can define angular acceleration as the rate of change of angular velocity. So, the formula becomes


  Angular Acceleration (α) = Angular Velocity (ω) / time (t)

Also, we can define Angular Velocity as the rate of change of angle θ. Hence, 


  Angular Velocity (ω) = Angle (θ) / time (t)

Substituting the value of ω in the first equation, we get


  Angular Acceleration (α) = (Angle / time) / time

                         α = Angle / time2

                         α = Angle * time-2

In physics, Angle (θ) is a dimensionless quantity, and we use [T] to represent Time. Hence, the equation becomes


  α = [T-2]

Since the formula doesn’t include the fundamental quantities Mass and Length, we can use [M0] * [L0], which is equal to 1.


  Angular Acceleration = [M0] * [L0] * [T-2]

Hence, the dimensional equation of angular acceleration is α = [M0 L0 T-2].

Derivation of Dimensional Formula of Angular Acceleration

2. What is the dimensional formula of acceleration due to gravity?

The dimensional formula of acceleration due to gravity is

[ M0 L1 T-2 ]

where [M], [L], and [T] are the fundamental quantities: Mass, Length, and Time.

Derivation

In physics, we can define acceleration due to gravity as the acceleration of a free-falling body due to gravitational force.

We know the formula of Force,


  Force = mass * acceleration due to gravity (g)

  g = Force / mass

The dimensional formula of Force is [ M1 L1 T-2 ], and we represent mass with [M]. Hence, the formula becomes


  g = [ M1 L1 T-2 ] / [M1]

  Acceleration due to gravity = [ M0 L1 T-2 ]

Hence, the dimensional equation of acceleration due to gravity is g = [ M0 L1 T-2 ].


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