What Are the Dimensions of Linear Momentum?

Linear momentum is a fundamental concept in mechanics, describing the state of motion of a body quantitatively as the product of its mass and velocity. Understanding the dimensions of linear momentum is essential for unit analysis, equation verification, and problem-solving in physics.

Definition and Physical Significance of Linear Momentum

Linear momentum, denoted by $p$, is defined as the product of a body’s mass ($m$) and its velocity ($v$). It is a vector quantity, indicating both magnitude and direction.

The formula for linear momentum is $p = m \times v$. This concept is central to the laws of motion and is conserved in isolated systems.

Derivation of the Dimensional Formula for Linear Momentum

To find the dimensional formula, consider the fundamental quantities involved in linear momentum. Mass has the dimension $[M]$, and velocity has the dimension $[LT^{-1}]$.

Using the formula $p = m \times v$, substitute the dimensions of mass and velocity:

$\begin{aligned} \text{Linear Momentum} &= \text{Mass} \times \text{Velocity} \\ &= [M] \times [LT^{-1}] \\ &= [MLT^{-1}] \end{aligned}$

Thus, the dimensional formula for linear momentum is $[MLT^{-1}]$.

For more details on related dimensional formulas, refer to the Dimensions Of Momentum page.

Units of Linear Momentum in SI and CGS Systems

In the SI system, the unit of linear momentum is kilogram metre per second ($\text{kg} \cdot \text{m} \cdot \text{s}^{-1}$). In the CGS system, it is gram centimetre per second ($\text{g} \cdot \text{cm} \cdot \text{s}^{-1}$).

Comparing Linear Momentum with Related Quantities

Linear momentum and impulse share the same dimensional formula, $[MLT^{-1}]$, as impulse is the product of force and time. However, force has an additional negative exponent for time, resulting in $[MLT^{-2}]$. Angular momentum introduces an extra power of length due to its rotational nature.

Understanding these differences is key for distinction in problems involving force, momentum, and rotational motion. For further study on related dimensions, see Dimensions Of Force.

Dimensional Analysis: Linear Momentum in Terms of Velocity, Density, and Frequency

Linear momentum can also be expressed in terms of other quantities for advanced dimensional analysis. Consider expressing its dimensions in terms of velocity $(v)$, density $(\rho)$, and frequency $(\nu)$.

Let $p \propto v^A \cdot \rho^B \cdot \nu^C$. The dimensions are as follows: velocity $[LT^{-1}]$, density $[ML^{-3}]$, and frequency $[T^{-1}]$.

Equating dimensions: $[MLT^{-1}] = [L^A T^{-A}] [M^B L^{-3B}] [T^{-C}]$

Solving for powers $A, B, C$, we find $A=4$, $B=1$, $C=-3$. Therefore, linear momentum in these terms is $p \propto v^4 \rho \nu^{-3}$.

This approach is frequently used in dimensional analysis, especially in advanced JEE problems. To explore related concepts, visit Dimensions Of Velocity.

Applications of the Dimensional Formula of Linear Momentum

The dimensional formula of linear momentum is essential for checking the consistency of physical equations, converting units, and solving conservation of momentum problems. It is also used when validating the equations in mechanics and other branches.

In collision and impulse problems, confirming that both sides of an equation have the dimension $[MLT^{-1}]$ is a fundamental step.

Sample Example: Using Dimensions of Linear Momentum

If a force $F$ acts on a body for a time $t$, the impulse delivered is $F \times t$. The dimensional formula is $[MLT^{-2}] \times [T] = [MLT^{-1}]$, matching that of linear momentum. This confirms impulse and linear momentum are dimensionally consistent.

Similar exercises are frequently tested in competitive exams. Reviewing practice questions aids in mastering these applications. See also Dimensions Of Acceleration for related calculations.

Common Mistakes and Important Points

• Always use SI units unless specified otherwise
• Do not confuse angular and linear momentum formulas
• Retain the exponent “$-1$” for time
• Check for dimensional consistency in equations

Careful attention to these points helps avoid common errors in exam settings, especially in calculations involving momentum, force, or impulse.

Summary Table: Dimensions of Linear Momentum and Related Quantities

A systematic understanding of dimension analysis strengthens foundational concepts and problem-solving skills in competitive exams. For additional resources, review the Dimensions Of Energy section for further practice.