Dimension of Momentum : Definition, Formula and Derivation

Dimension of Momentum Overview

Momentum is a fundamental concept in physics that refers to the measure of an object's motion. It is defined as the product of an object's mass and velocity and is a vector quantity, meaning it has both magnitude and direction. The dimension of momentum is usually represented as kilogram meters per second (kg m/s), which is the standard unit of measurement for momentum in the SI (International System of Units).

Dimension of Momentum: What is Momentum?

Momentum is defined as the result of multiplying the mass of a particle by its velocity.

It is a vector quantity, possessing both direction and magnitude. Newton's second law states that the rate of change in momentum is equal to the force acting on a particle. This means that if a constant force is applied to a particle over a certain period of time, the product of that force and time will equal the change in momentum. Additionally, the momentum of a particle can be thought of as a measure of the time it takes for a constant force to bring the particle to rest. If either the mass or velocity of a particle is zero, its momentum will also be zero.

Mathematically, the formula is:

P = m x v

Here,

• p represents momentum
• m represents mass
• v represents velocity

What is Dimension of Momentum?

The dimension of momentum is represented as [M] and is given by the product of the dimensions of mass and velocity. In the SI (International System of Units), mass is measured in kilograms (kg) and velocity is measured in meters per second (m/s). Therefore, the dimensional formula for momentum is given as:

[momentum] = [mass] x [velocity] = kg x m/s

So, the dimension of momentum is [M] = kg x m/s. This formula shows that momentum has the dimensions of mass times velocity and is a vector quantity with both magnitude and direction. The magnitude of momentum is proportional to an object's mass and velocity, and its direction is the same as the velocity vector.

The 3 Dimension of Momentum

Momentum is a vector quantity that describes the motion of an object in three dimensions:

1. Magnitude: The magnitude of momentum is the product of an object's mass and velocity. It is a scalar quantity and is usually represented by the letter "p." The greater the mass and velocity of an object, the greater its magnitude of momentum.
2. Direction: Momentum is a vector quantity and thus has a direction as well as a magnitude. The direction of momentum is determined by the direction of motion of the object. If an object is moving to the right, its momentum will be in the positive x-direction, whereas if it is moving to the left, its momentum will be in the negative x-direction.
3. Plane of motion: Momentum can also exist in three dimensions, and not just in a single plane of motion. This means that an object can have momentum in the x, y, and z-directions, depending on its direction of motion. For example, a baseball that is hit into the air has momentum in both the x and y-directions, as it is moving forward and upward at the same time.

Dimension of Momentum: Types

The three types of dimensions of momentum are:

1. Scalar Momentum: Scalar momentum, also known as magnitude of momentum, is a scalar quantity that describes the strength of the momentum of an object. It is determined by the product of the object's mass and velocity, and is denoted by the symbol "p". The magnitude of momentum does not specify the direction of motion, only the strength of it.
2. Linear Momentum: Linear momentum, also known as vector momentum, is a vector quantity that describes the direction and magnitude of an object's motion. It is denoted by the symbol "p" and is given by the equation p = mv, where "m" is the object's mass and "v" is its velocity. Linear momentum is a measure of the object's tendency to continue moving in a straight line.
3. Angular Momentum: Angular momentum is a measure of an object's rotational motion about a fixed axis. It is a vector quantity and is denoted by the symbol "L". It is given by the equation L = r x p, where "r" is the position vector of the object and "p" is its linear momentum. Angular momentum is important in understanding the behavior of rotating objects, such as spinning tops and planets.

These different types of momentum describe different aspects of an object's motion and are essential for understanding the physics of motion and mechanics.

Also Read About- Linear Dimensional Formula of Momentum

Dimension of Momentum Example

The dimension of momentum can be demonstrated with an example. Let's consider a simple scenario where we have a ball with a mass of 0.1 kilograms (kg) and a velocity of 10 meters per second (m/s).

The momentum of the ball can be calculated using the equation for momentum:

p = mv = 0.1 kg x 10 m/s = 1 kg m/s

So, the momentum of the ball is 1 kilogram meter per second (kg m/s). The units for momentum are the product of the units for mass and velocity.

In this example, we can see that the dimension of momentum is the same as the product of the dimensions of mass and velocity. This formula helps us understand the physical meaning of momentum and how it is related to the mass and velocity of an object.

Derivation of Unit of Momentum

The momentum of a certain object is represented by the symbol ‘p'.

The mathematical equation of momentum of the object is : p = m × v

Where,

• m = mass of a particle
• V = velocity of the particle

The equation above describes the fact that the mass and velocity of the particle are directly proportional to its momentum.

Hence, the SI unit that we get of the momentum of a body/ particle is the product of its mass (Kg) and its velocity (ms-1). Hence the SI unit of momentum can be written as kg ms-1.

For two particles, the momenta can be added in the following manner:

P = p1 + p2

= m1 v1 + m2 v2

In the case of more than two particles, the momentum will be added in the following way: 

p = imi vi

Dimension of Momentum: Other Units

Considering a situation in which the force is equal to the rate of change of momentum:

Force = Change in Momentum/Time Interval

Thus, Change in momentum = Force × time interval

Hence, the unit for momentum here can be written as Newton-second (Ns).

According to the CGS system, if we calculate the velocitobject/s and mass in grams, then the SI unit of momentum can be written as gram-centimeters per second i.e. (g.cm/s)

There are even a few more units of momentum, like kg×mi/hr, kg×km/hr and g×cm/s. In every case, the unit of the mass of the object is multiplied by the units of velocity.

Momentum Units

The units of momentum are listed below:

Importance of Dimension of Momentum in IIT JEE Exam

It is critical to take a holistic approach to every facet of a subject's chapter. It will not only adequately prepare you for the exam but will also clarify your understanding of each topic. It will help you in IIT preparation and answer conceptual problems in the exam. The number of questions from the chapter unit and dimensions would be one or two, with a weightage of roughly four marks.

Dimension of Momentum: Things to Remember

• The momentum itself does not have any SI unit. Hence, the product of the SI units of mass & velocity is used for the SI unit of momentum which is kg ms-1.
• The mathematical formula used to describe momentum is p = m x v.
• The direction of the velocity vector is the same as the direction of the momentum vector.
• If we consider the launch of a rocket, the initial momentum is zero. During the launch due to the exhaustion of expanding gases the downward momentum gets equalized with upward momentum. Hence, the total momentum remains constant i.e. zero.
• Momentum is a vector quantity which involves both the direction and the magnitude.
• Considering Newton's Third law, all particles exerts equal and opposite forces on each other. So, the change in momentum in one particle will be balanced by some other particle with an opposite and equal change in momentum.
• The law of conservation of momentum is applied to all the interactions including the separation and collisions that are caused by explosive forces.

Related Articles-