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14 Vector Sum Find the sum of the three vectors shown in Figure 5-7 15 Commutative Operations The order in which vectors are added does not matter Mathematicians say that vector addition is commutative Which ordinary arithmetic operations are commutative Which are not 16 Critical Thinking A box is moved through one displacement and then through a second displacement The magnitudes of the two displacements are unequal Could the displacements have directions such that the resultant displacement is zero Suppose the box was moved through three displacements of unequal magnitude Could the resultant displacement be zero Support your conclusion with a diagram

Remember that because this plug-in also uses plug-ins 21 and 22, PIPHP_RelToAbsURL() and PIPHP_GetLinksFromURL(), you should also copy and paste them into your program, or otherwise include them

Figure 5-7

13 Components Find the components of vector M, shown in Figure 5-7

ush your hand across your desktop and feel the force called friction opposing the motion Push your book across the desk When you stop pushing, the book will continue moving for a little while, then it will slow down and stop The frictional force acting on the book gave it an acceleration in the direction opposite to the one in which it was moving So far, you have neglected friction in solving problems, but friction is all around you You need it to both start and stop a bicycle and a car If you have ever walked on ice, you understand the importance of friction

This is a simple calculated measure It s not really much of a calculation but just shows how to alias a measure

.

function PIPHP_CheckLinks($url, $links) { $results = PIPHP_GetLinksFromURL($url); $missing = array(); $failed = 0; foreach($links as $link) if (!in_array($link, $results)) $missing[$failed++] = $link; if ($failed == 0) return array(TRUE); else return array(FALSE, $missing); }

There are two types of friction Both always oppose motion When you pushed your book across the desk, it experienced a type of friction that acts on moving objects This force is known as kinetic friction, and it is exerted on one surface by another when the two surfaces rub against each other because one or both of them are moving To understand the other kind of friction, imagine trying to push a heavy couch across the floor You give it a push, but it does not move Because it does not move, Newton s laws tell you that there must be a second horizontal force acting on the couch, one that opposes your force and is equal in size This force is static friction, which is the force exerted on one surface by another when there is no motion between the two surfaces You might push harder and harder, as shown in Figures 5-8a and 5-8b, but if the couch still does not move, the force of friction must be getting larger This is because the static friction force acts in response to other forces Finally, when you push hard enough, as shown in Figure 5-8c, the couch will begin to move Evidently, there is a limit to how large the static friction force can be Once your force is greater than this maximum static friction, the couch begins moving and kinetic friction begins to act on it instead of static friction A model for friction forces On what does a frictional force depend The materials that the surfaces are made of play a role For example, there is more friction between skis and concrete than there is between skis and snow It may seem reasonable to think that the force of friction also might depend on either the surface area in contact or the speed of the motion, but experiments have shown that this is not true The normal force between the two objects does matter, however The harder one object is pushed against the other, the greater the force of friction that results.

Figure 5-8 There is a limit to the ability of the static friction force to match the applied force