# Linear impulse-momentum relations Notes pdf ppt

**Definition of the linear impulse of a force:**

In most dynamic problems, particles are subjected to forces that vary with time. We can write this mathematically by saying that the force is a vector valued function of time .

If we express, the force as components in a fixed basis , then

where each component of the force is a function of time.

** **

The ** Linear Impulse **exerted by a force during a time interval is defined as

The linear impulse is a vector, and can be expressed as components in a basis

If you know the force as a function of time, you can calculate its impulse using simple calculus. For example:

- For a
*constant*force, with vector value the impulse is - For a
*harmonic*force of the form the impulse is

It is rather rare in practice to have to calculate the impulse of a force from its time variation.

**Definition of the linear momentum of a particle:**

The linear momentum of a particle is simply the product of its mass and velocity

The linear momentum is a vector its direction is parallel to the velocity of the particle.

** Impulse-momentum relations for a single particle:**

- Consider a particle that is subjected to a force for a time interval .
- Let denote the impulse exerted by
**F**on the particle - Let denote the change in linear momentum during the time interval .

The momentum conservation equation can be expressed either in differential or integral form.

In differential form:

In integral form:

This is the integral of Newton’s law of motion with respect to time.

The impulse-momentum relations for a single particle are useful if you need to calculate the change in velocity of an object that is subjected to a prescribed force

**Impulse-momentum relation for a system of particles:**

Suppose we are interested in calculating the motion of several particles, sketched in the figure.

**Total external impulse on a system of particles: **

Each particle in the system can experience forces applied by:

** Other particles in the system** (e.g. due to gravity, electric charges on the particles, or because the particles are physically connected through springs, or because the particles collide). We call these

**acting in the system. We will denote the internal force exerted by the**

*internal forces**i*th particle on the

*j*th particle by .

Note that, because every action has an equal and opposite reaction, the force exerted on the *j*th particle by the *i*th particle must be equal and opposite, to , i.e. .

** Forces exerted on the particles by the outside world **(e.g. by externally applied gravitational or electromagnetic fields, or because the particles are connected to the outside world through mechanical linkages or springs). We call these

**acting on the system, and we will denote the external force on the**

*external forces**i*th particle by

We define the *total impulse* exerted on the system during a time interval as the sum of all the impulses on all the particles. It’s easy to see that the total impulse due to the internal forces is zero because the *i*th and *j*th particles must exert equal and opposite impulses on one another, and when you add them up they cancel out. So the total impulse on the system is simply

** **

**Total linear momentum of a system of particles: **

The total linear momentum of a system of particles is simply the sum of the momenta of all the particles, i.e.

The impulse-momentum equation can be expressed either in differential or integral form, just as for a single particle:

- In differential form

2. In integral form

This is the integral of Newton’s law of motion with respect to time.

**Conservation of momentum: **

If *no* external forces act on a system of particles, *their total linear momentum is conserved, *i.e. .