Write the dimensional formula of force.

Short Answer

Force is a physical quantity that causes an object to move or change its motion. According to Newton’s second law, force is equal to mass multiplied by acceleration. To find its dimensional formula, we express both mass and acceleration in terms of basic dimensions.

Mass has the dimension M, and acceleration has the dimension L T⁻². Therefore, the dimensional formula of force becomes M L T⁻². This formula helps us understand the nature of force and is used in many calculations and dimensional analysis methods in physics.

Detailed Explanation :

Dimensional Formula of Force

Force is one of the most important quantities in physics. It plays a central role in classical mechanics and explains why objects start moving, stop moving, or change direction. To express force in a scientific and standard way, we use the concept of dimensions. Dimensions help us understand what basic physical quantities combine to form more complex quantities like force.

The dimensional formula tells us the fundamental nature of a physical quantity in terms of mass (M), length (L), and time (T). By writing the dimensional formula of force, we can use it for checking equations, deriving formulas, and converting units. Understanding the dimensional form makes learning physics simple, logical, and systematic.

Definition of Force

Force is defined as any influence that changes or tends to change the state of motion of an object. Newton’s second law gives the mathematical form of force:

Force (F) = Mass (m) × Acceleration (a)

This equation becomes the base for deriving the dimensional formula of force. Since both mass and acceleration have their own dimensional representations, we can combine them to get the dimension of force.

Dimensions Involved in Force

Before writing the complete dimensional formula, we must understand the dimensions of the components of force:

  1. Dimension of Mass (m)

Mass is a fundamental physical quantity.
Its dimension is:
[M]

  1. Dimension of Acceleration (a)

Acceleration is defined as the rate of change of velocity with respect to time.
Velocity = displacement/time = L T⁻¹
Acceleration = velocity/time = (L T⁻¹) / T = L T⁻²

So, the dimension of acceleration is:
[L T⁻²]

Deriving the Dimensional Formula of Force

Using the equation:

Force = Mass × Acceleration

Substitute the dimensions of mass and acceleration:
Force = [M] × [L T⁻²]

Now multiply them:
Force = [M¹ L¹ T⁻²]

Thus, the dimensional formula of force is:

[F] = M L T⁻²

This formula shows that force depends on mass, length, and time. None of the dimensions are zero, indicating force is a derived quantity formed from fundamental quantities.

Importance of Dimensional Formula of Force

Understanding the dimensional formula of force helps in many ways:

  1. Checking the validity of equations

If a formula involving force appears in a problem, we can verify if the equation is dimensionally correct by comparing both sides. If dimensions match, the formula is likely correct.

  1. Deriving unknown formulas

Sometimes, physics problems require deriving relationships using dimensions. The known dimensional formula of force helps in calculating quantities such as work, pressure, momentum, and energy.

  1. Converting units

Force is measured in newtons (N) in the SI system.
1 Newton = 1 kg × m/s²
This matches exactly with the dimensional form M L T⁻². Understanding dimensions makes converting units easier and more accurate.

Application in Other Physical Quantities

Many physical quantities are derived using force. Their dimensional formulas depend directly on the dimensional formula of force:

  • Work = Force × Distance
    = (M L T⁻²) × L
    = M L² T⁻²
  • Pressure = Force / Area
    = (M L T⁻²) / L²
    = M L⁻¹ T⁻²
  • Momentum = Mass × Velocity
    = M × (L T⁻¹)
    = M L T⁻¹

These examples show how force helps build other important physical concepts.

Why Dimensional Formula Helps Students

Dimensional formulas are useful not only for scientists but also for students. They make physics easier because:

  • They help students avoid memorizing too many formulas.
  • They show the logical connection between quantities.
  • They help in solving numerical problems quickly.
  • They allow checking for mistakes in formulas.

Therefore, the dimensional formula of force is simple but extremely useful for learning physics effectively.

Conclusion

The dimensional formula of force is derived using the basic relation F = m × a. Mass has the dimension M, and acceleration has the dimension L T⁻². Therefore, the dimensional formula of force becomes [F] = M L T⁻². This representation helps in checking equations, converting units, and understanding physical relationships. Knowing the dimensional form of force builds a strong foundation in physics and supports further learning of complex topics.