Ó 2000, W. E. Haisler Conservation of Linear Momentum 4

Ó 2000, W. E. Haisler Conservation of Linear Momentum 4

ENGR 211

Principles of Engineering I

(Conservation Principles in Engineering Mechanics)

Conservation of Linear Momentum

·  Introduction

·  Conservation of Linear Momentum Equations

·  Linear Momentum and Newton's Laws of Motion

·  Steady State

·  Other Special Cases

·  Reference Frame

Introduction

The linear momentum of a system is defined to be the product of the mass (m) and velocity () of the system. Since velocity is a vector, then linear momentum is a vector quantity (has three components). Because linear momentum is a conserved property, there is no generation or consumption. Therefore, the conservation statement becomes

Linear Momentum Possessed by Mass

·  Linear Momentum may enter or leave the system with mass.

·  Linear Momentum may also enter or leave the system at a certain rate.

·  When mass enters a system traveling at a velocity (), it adds momentum to the system.

·  The rate at which mass (m) enters the system is (where ) and therefore the rate at which linear momentum enters the system is given by . Similarly, rate at which the linear momentum leaves the system is .

·  The proper way to look a momentum rate is as follows

If , then the above can be written in equation form:

·  If there are multiple masses entering or leaving the system (say n of them), then we simply add them up:

and

·  Note that the velocity of a mass can be thought of as it's linear momentum per mass.

Linear Momentum in Transit: Forces

·  Why do we refer to "Forces" as "Linear Momentum in Transit?" Newton's 2nd Laws states that the external forces on a system are connected to the time rate of change of linear momentum within the system!

·  Linear momentum enters the system as the result of forces that act upon the mass in the system. By Newton's second law, an external force acting on a system provides momentum to the system at rate . These forces may act as surface or contact forces on the system boundary (called tractions), or they may act as body forces on the entire mass of the system (for example, body forces due to gravity).
Schematic for Conservation of Linear Momentum

·  The external forces may be due to:

·  pressure acting over the surface area of the system

·  drag or frictional forces acting on the surface

·  supporting structure which holds the system in place (cause forces acting on the surface)

·  gravitational and electrostatic body forces acting on the system volume

Conservation of Linear Momentum Equations

We can now state the rate equation for conservation of linear momentum as:

The momentum of the system may change with time because of mass that enters and leaves, and because of external forces that act on the system.

As mass enters the system, the amount of linear momentum that enters the system is this [mass flow rate] times the [momentum per unit mass (i.e., the velocity) of the mass].

We must keep in mind that conservation of linear momentum is a vector equation! Consequently, we can write in component form. For a Cartesian coordinate system, we have

x-direction:

y-direction:

z-direction:

Linear Momentum of the System

The linear momentum of the system may be written (for n particles in the system) as

Rate of Change of Linear Momentum of the System

·  Consider the time rate of change of linear momentum of the system (left side of conservation equation) and use the product rule for the derivative:

·  If the mass of the system is constant with time, then the time rate of change of linear momentum reduces to

Note that the time derivative of the velocity becomes the acceleration:

Velocity of the Center of Mass

For rigid bodies and other closed systems, it is convenient to define a velocity that characterizes the system as a whole. This velocity is an average velocity for the system and acts at the center of mass. We define the velocity of the center of mass, , such that the system mass times the velocity of the mass center is equal to system linear momentum:

and solving for the velocity of the center of mass:

If the size of the elements mi becomes differentially small, the summation process becomes an integration and the velocity of the center of mass becomes

This is sometimes called the mass-average velocity of the system.
Recall that the velocity of an element of mass is defined to be the time rate of change of the position vector describing it's location. For a particle with velocity whose position is refined by the position vector :

, x=x(t), etc.

The center of mass can be determined in terms of the position vectors of all of the mass elements of the system:

As the size of mass particle becomes differentially small, we can replace the summation by an integration to obtain

This definition of the center of mass applies only for a closed system (one in which there is no mass into or out of the system). When the system is closed (no mass in or out of system), the conservation of linear momentum becomes

Integral (Finite Time Period) Equation

The rate form of the conservation of linear momentum equation can be integrated over a finite time period to obtain

The above equation can also be written for a finite time period as

or

Linear Momentum and Newton's Laws of Motion

The concept of conservation of linear momentum applied to a rigid body, together with the realization that forces exchange momentum between the surroundings and the body, is actually a statement of Newton's three laws of motion.

·  Newton's first law of motion - A body which is at rest will stay at rest unless acted upon by an external force. In other words, a body does not spontaneously generate momentum; momentum is conserved.

·  Newton's second law of motion - For a body (system) of constant mass, the sum of the external forces is equal to the mass times acceleration of the body. This is clearly a result of the conservation of linear momentum when so that by conservation of mass

·  Newton's third law of motion - For every force there is an opposite and equal reaction force. This also implies conservation of linear momentum. A force acting on a system by the surroundings (boundary forces or body forces) provides an opposite and equal force exerted on the surroundings. Thus, momentum transferred to the system must have been provided or given up by the surroundings, implying conservation of linear momentum.

Steady State

The steady state condition for a system means that the system is not changing with time and hence the accumulation within the system is zero for any time period.

For a finite time period:

or, for any instant of time:

Through conservation of linear momentum, steady state implies:

Steady state also implies that the term in the conservation of mass law.

For rigid body statics (no mass enters or leaves the system and the body is at steady state), linear momentum reduces to

Reference Frame

In order to define velocity and liner momentum, we must define a reference coordinate system. Position and velocity are then measured with respect to the origin of the coordinate system. For this purpose, we can choose any reference frame so long as it is an inertial reference frame, i.e., one that is not accelerating (has constant velocity and not rotating).

Consider a block sliding along a plane at a velocity of vref with an attached pendulum as shown to the right. The x-y frame is fixed. The x'-y' frame is attached to the block so that it also has a velocity of vref.

Now lets write COLM with respect to the stationary x-y reference frame

(x,y coords.) (1)

Substitute the relation between the two coordinates systems into (1):

Taking the time derivative on the left side and rearranging, the above can be written:

The first underlined term is zero since is constant. The double underlined term since this is conservation of mass (for no mass gen/con). Thus, the COLM equation in the x'-y' frame reduces to

(x',y' coords.) (2)

Note that equation (2) is identical to equation (1) except for the change in reference frame. Since (2) was derived from (1), we see that COLM is independent of coordinate system as long as we use an inertial reference frame for the velocities and the external forces are independent of the reference frame.

Some notes on unit and radius vectors. Consider a line element OP of length L (or a vector ) where . The vector is given by . Note: "O" is at the origin.

We define the following angles:

Now the cosine of each angle is: , , and . Note that is an angle in the plane containing the x-axis and the line segment OP. A unit vector in the direction of is given by

where . The cosines of the three angles are often referred to as the direction cosines of the line OP.

From geometry and trigonometry, one can prove that . Thus you only need to specify two of the angles – the third must satisfy the trig relation. This is similar to a unit vector – two of the components (say x and y) can be specified, but the third (say z) must satisfy !!

A line segment AB of length L (or a vector) may also be located by the coordinates of its end points: A (xbeg, ybeg, zbeg) and B (xend,yend,zend). Then the unit vector in the direction of AB is given by:

where , etc.

Example: Vectors and in x-y plane:

Note on conversion of and in American Engineering units.

Note units:

Notes on determining the mass entering a system. The mass entering a system boundary (through an area) during a period of time may be determined or specified in many ways.

1.  We can simple state that a certain amount of mass enters the system during a specified time period, i.e., .

2.  However, a more common situation is when one specifies that a fluid (of given density) flows through an area at a specified velocity normal to the area. In this case, consider the following dimensional analysis:

So the rate of mass entering a system for a mass with density r and passing through an area A with a velocity Vn normal to the area is given by: .

3.  Another common way to describe mass flow is by the mass flux rate, or the mass per unit area per unit time.

.

To obtain the mass entering a system (in terms of mass flux rate) we write: