Kinetics Of A Particle: Impulse & Momentum

Principle Of Linear Impulse And Momentum:

The basis of principle of linear impulse and momentum is given by:

The above equation can be written as:

 The above equation is called Principle of Linear and Momentum.

• Impulse measures the effect of force during small time period 'dt'.
• Momentum (is the strength of motion) is associated with the states of process.
• Impulse are associated with process caused by external force.

For System Of Particles:

For system of particles, the principle of linear impulse and momentum is written as the summation for number of particles:

Here,

• Individual Particle Velocity (Vi) = Velocity of center of mass (Vg).
• Summation Of individual particle (mi) = Total mass of body (m).

Conservation Of Linear Momentum For System Of Particles:

If the sum of external impulse of the system of particles is zero, then Principle of linear momentum and impulse becomes:

There are two types of force on the basis of Impulse which are described ahead:

• Non-Impulsive Force  →  If forces produce negligible amount of impulse.
• Impulsive Force  →  If force produce significant amount of impulse due to the striking of one body against other. Impulsive force require Impulse Momentum Analysis.

Impact:

It refers to the collision between bodies characterized by large contact forces in very short time interval.

Impact is further divided into two types which are:

• Central Impact  →  If the colliding motion of two particles is along the line of impact.
• Oblique Impact  →  If the direction of one or both particles is at some angle with the line of impact.

Analysis Of Central Impact:

Consider particles as deformable or non-rigid bodies which pass through several stages which are described as:

There are two types of impulse which we should know for proper understanding of this topic.

1. Deformation Impulse  →  The impulse which is applied on bodies during collision and can deform other body.
2. Restitution Impulse  →  The reaction impulse which is applied to remove bodies in contact during collision.

Always keep in mind that deformation impulse is greater than deformation impulse i.e ∫ P dt  > ∫ R dt.

Applying Principle of Linear Momentum and Impulse, we get:

For body A, Separating the deformation Impulse:

There are two cases for impact which are:

• Elastic Impact  →  If coefficient of restitution is 1 (e = 1) that means deformation impulse is equal to the restitution impulse.
• Plastic Impact  →  If coefficient of restitution is 0 (e = 0) that means restitution impulse is zero and bodies stick to each other.

Analysis Of Oblique Impact:

Momentum of the system along X-axis is conserved to applying conservation of linear momentum

And, applying principle of impulse and linear momentum along Y-axis.

The coefficient of Restitution is given by:

Angular Momentum:

Angular momentum is defined as: "Moment of linear momentum". In Scalar formulation:

In vector formulation: Angular momentum is the cross product of position vector and momentum. 

Relation Between Moment Of A Force & Angular Momentum:

The moment of a force is defined as the cross product of position vector and force. Mathematically:

Principle Of Angular Impulse & Momentum:

According to the relation between angular momentum and moment of a force:

After integration:

The above equation can be written as:

And,

Conservation Of Angular Momentum:

If there is no angular impulse between bodies, then:

• How impulse and momentum are measured? See the video given below:

References:

• Material from Class Lectures + Book named Engineering Mechanics Statics & Dynamics by R.C. Hibbeler (12th Edition) + from YouTube channel named Yiheng Wang for Engineering Dynamics Mechanics + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.