Study Guides/Physics/Dimension of g (Gravity)
Study Guide · Physics

What is the Dimensional Formula of 'g' (Acceleration Due to Gravity)?

In physics, there is a massive difference between the small letter 'g' and the capital letter 'G'. While 'G' is the Universal Gravitational Constant, the small 'g' stands for the Acceleration Due to Gravity (the rate at which an object speeds up as it falls towards the Earth).

The dimensional formula of 'g' is [M⁰ L¹ T⁻²].

Question (Click to Flip)

What is the dimensional formula of small g?

Answer

The dimensional formula for acceleration due to gravity (g) is [M⁰ L¹ T⁻²].

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Key Facts

Quantity: Acceleration due to Gravity (small 'g').

Dimensional Formula: [M⁰ L¹ T⁻²].

SI Unit: meters per second squared (m/s²).

Value on Earth: Approximately 9.8 m/s².

Crucial Rule: Small 'g' is an acceleration, it is NOT a Force. (Force of gravity is Weight = mg).

Step-by-Step Derivation

Even though gravity is a mysterious force, small 'g' is simply a type of Acceleration. Therefore, its dimensional formula is exactly identical to the standard formula for acceleration.

Let's break it down:

  1. The basic formula for acceleration is: Change in Velocity / Time taken.
  2. Velocity is: Distance / Time.
  3. Putting it together: Acceleration = Distance / (Time × Time).

Now, let's assign the fundamental dimensions:

  • Distance = Length [L].
  • Time = [T].
  • Therefore: [L] / [T²]
  • Moving T to the numerator: [L¹ T⁻²].

Adding the Mass Element

To write it in the standard [M L T] format, we must include Mass. Since the acceleration of a falling object does NOT depend on its mass (a feather and a bowling ball fall at the same speed in a vacuum), the Mass dimension is zero.

Final Dimensional Formula: [M⁰ L¹ T⁻²].

The SI Unit

Looking at the dimension [L¹ T⁻²], we can easily derive the official SI unit for small 'g'.

  • [L] = meters (m)
  • [T⁻²] = per second squared (/s²) Therefore, the SI unit of 'g' is m/s². (On the surface of the Earth, its value is 9.8 m/s²).

Questions and Answers

What is the dimensional formula of small g?+

The dimensional formula for acceleration due to gravity (g) is [M⁰ L¹ T⁻²].

Why is the dimension of mass zero in the formula?+

Because the acceleration of a falling object is completely independent of its mass. Therefore, the mass dimension [M] is raised to the power of zero.

Is the dimension of small g different from capital G?+

Yes, they are completely different. Small 'g' is an acceleration [L T⁻²], while capital 'G' is the Universal Gravitational Constant [M⁻¹ L³ T⁻²].

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