Download presentation

Presentation is loading. Please wait.

Published byKacie Nemo Modified over 2 years ago

1
Cht. IV The Equation of Motion Prof. Alison Bridger 10/2011

2
Overview n In this chapter we will: n Develop - using Newton’s 2nd Law of Motion - an equation that in principal - will enable us to predict flows. n Identify the forces which cause air motions. n Adapt the resulting equation to Earth. n See how the various forces work. n See what the resulting equation(s) look like in component form.

3
Introduction n In the previous chapter, a flow was assumed. n In this chapter we will develop an equation to predict the flow of air n This is called the Momentum Equation, or the Equation of Motion.

4
Introduction n In the next chapter, we’ll begin the process of solving this equation. n NOTE: flow evolution also depends on thermodynamic variables. n For example, we know (MET 61) that flows are forced by pressure gradients which follow from heating variations.

5
Introduction n In other chapter of CAR we will develop additional equations that forecast the entire evolution of the flow - dynamic and thermodynamic (via the variables V, p, T, etc.). n The entire set of equations are called the Equations of Motion.

6
Newton’s 2nd Law of Motion n Read the full definition on p. 142. n We have:

7
Newton’s 2nd Law of Motion

8
n We sum over all possible forces (“F i ”) per unit mass. n If we can solve – integrate over time – we gain knowledge of the future velocity of the air parcel. n Speed & direction.

9
Newton’s 2nd Law of Motion n Here are some steps we need to follow: u identify the forces that affect air parcel motions u formulate how these work (mathematically) u substitute these forms back into the Eqn. above (“F=ma”), and then...

10
Newton’s 2nd Law of Motion n There is a problem (of course!) n Earth is rotating - this makes it a non-inertial frame of reference. n However, Newton’s 2nd Law is formulated for an inertial frame of reference.

11
Newton’s 2nd Law of Motion n Thus we must “adapt” the equation that we have developed to a non-inertial frame of reference. n This introduces the Coriolis force!

12
Forces... n We can start by identifying 2 basic types of forces: u body forces… F affect the entire body of the fluid (not just the surface) F act at a distance

13
Forces... F examples are F gravitational forces (not the same as gravity!) F magnetic forces F electrical forces F we ignore the latter two (lower atmosphere only!)

14
Forces... u surface forces… F affect the surface of a fluid parcel F caused by contact between fluid parcel examples are pressure forces viscous forces (friction)

15
Gravitation n Newton’s Law of Gravitation…p.137 n the magnitude of the force is given by u G a = GmM / r 2 u G is the Universal Gravitation Constant u m and M are the two masses u r is the distance between the two centers of masses

16
Gravitation n We assume M = Earth’s mass (so M M e ) and m = mass of air parcel (we will soon set m=1). n For the force direction, we assume: u Earth is a sphere u Earth is not rotating u Earth is homogeneous (so the center of mass is at the Earth’s geometric center) n As a result, we can write:

17
Gravitation

18
n For the case m=1 (unit mass), G a g a and we have:

19
Gravitation

20
n Here, G e is Earth’s Gravitation Constant (=GM e ). n And note that the lower case (g a ) means “per unit mass”. n So - the expression above for g a goes into the RHS of our expression of Newton’s 2nd Law (above).

21
Friction n This topic will be treated in detail in MET 130 etc. n In dynamics we typically take one of two approaches: u ignore friction - it’s usually a “second order” correction to the “important” “first-order” dynamics u assume something very simple for friction

22
Friction u example…we may set friction to depend linearly on the strength of the existing wind, as in

23
Friction

24
n Here, is a constant that determines the rate of decay of V with time due to friction. n When we wish to allow for friction in our work, we will often simply add “F” to our Eqn. Of Motion - you should remember that “F” stands for friction.

25
Pressure Gradient Force n The last force important in driving motions is the pressure gradient force. n In many respects, it is the most important force since it initiates motions! n It is important to remember that it is pressure gradients that matter - not actual pressures themselves.

26
Pressure Gradient Force n Fluid parcels experience pressure forces due to contact with surrounding parcels. n When these forces are spatially variable, the parcel will experience a net motion. n We need to quantify this...

Similar presentations

OK

Chapter 5 Force and Motion In Chapters 2 and 4 we have studied “kinematics,” i.e., we described the motion of objects using parameters such as the position.

Chapter 5 Force and Motion In Chapters 2 and 4 we have studied “kinematics,” i.e., we described the motion of objects using parameters such as the position.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on surfing the net safely Ppt on covering the environmental problems causes effects and solutions Ppt on layer 3 switching tutorial Ppt on advertisement Ppt on fdi retail in india Ppt on mohandas karamchand gandhi indians Ppt on scramjet engines Ppt on festivals of kerala Ppt on quality control in industrial management Best ppt on natural disasters