## General Introduction to Applications

Now that we

knowNewton's Laws of Motion, how do weapplythem? How can they let uspredictthe motion of an object if we know all the forces acting upon it? How can they let uspredictthe forces on an object if we know its motion?It is only a slight over statement to say that the rest of this course is just looking for devious ways to apply

F= ma

Example:Consider a crate being pulled along a frictionlessfloor (while such a floor isveryhard to find, this will still help us understand the conceptandwe can return to this situation later,afterconsidering friction, and solve it more realistically).Consider a crate being pulled along a horizontal, friction

lessfloor. A rope is tied around it and a man pulls on the rope with a force of T. T is thetensionin the rope. What happens to the crate?Before we can apply Newton's Second Law,

F= mawe must find the

netforce -- thevectorsum ofallthe forces -- acting on the object. In addition to the forceTexerted by the rope, what other forces acton the object?A good, clear diagram, showing

all the forcesis an important beginning foreveryMechanics problem. Sometimes you can draw the forces on top of a sketch of the situation. Sometimes you will need to draw the forces separately to make them clear. This is called a "free body diagram". As discussed in class, in Mechanics, we can restrict our attention to "contact forces" and "gravity". That means gravity pullsdownon this crate with a force equal to its weight,w. But the floor supports the crate. The floor responds by pushingupon the crate with a force we call thenormal force. "Normal" means "perpendicular". We will call this forcen; you may also encounter it labeledNorF_{N}.These forces are shown on the "free body diagram" above. We have drawn in

allthe forces actingonthe object. The net force is thevectorsumof these forces.F_{net}=F=T+n+wwhere , the Greek upper case "sigma", means "the sum of". Remember, tho', vector notation is

alwayselegant shorthand notation. When we writeF_{net}=F=T+n+wwe have really written

F _{net, x}= F_{x}= T_{x}+ n_{x}+ w_{x}

andF _{net,y}= F_{y}= T_{y}+ n_{y }+ w_{y}What are these x- and y-components of the forces

T, n,andw? For this first, simple example, we can find -- by inspection -- that these components areT _{x}= TT

_{y}= 0n

_{x}= 0n

_{y}= nw

_{x}= 0w

_{y}= - wwhere the minus sign means "down".

Now we are ready to apply Newton's Second Law,

F= maBut that must first be written in terms of components,

F _{x}= F_{net,x}= F_{x}= m a_{x}F

_{x}= F_{net,x}= F_{x}= T_{x}+ n_{x}+ w_{x}= T = m a_{x}T = m a

_{x}a

_{x}= T / mThe crate has a

horizontal accelerationequal to the tension T divided by m, the mass of the crate. What about the forces in the vertical direction?F _{y}= F_{net,y}= F_{y}= m a_{y}F

_{y}= F_{net,y}= F_{y}= T_{y}+ n_{y}+ w_{y}= n - w = m a_{y}n - w = m a

_{y}Since we know the crate does not accelerate in the y-direction -- it does not jump up off the floor and it does not burrow down into the floor -- we know a

_{y}= 0, son = w The upward normal force exerted by the floor on the crate, in this situation, is equal to the weight, the downward force of gravity. That will

notalways be the case but it is true in this example or in this situation.

Example:What are the forces acting on book if you pushdownon it with a forceFwhile it sits on a smooth, horizontal table as shown in the sketch below?

Draw in all the forces.This is called a "free body diagram". We will restrict ourselves, in this Mechanics section, to "contact forces" and the force of gravity. Contact forces, for this case, will be the "normal" force -- the perpendicular force -- exerted by the table -- labelednin the diagram -- and the forceFexerted by the hand. Gravity exerts a force downward, called the weight and labeledw. Just as in the previous example, we can immediately writeF= mabut that is really elegant shorthand for

F _{x}= F_{net,x}= F_{x}= m a_{x}

andF _{y}= F_{net,y}= F_{y}= m a_{y}In this example, tho',

nothing happens in the horizontal direction.All of the forces haveonlyvertical components so all we really have isF _{y}= F_{net,y}= F_{y}= m a_{y}Taking up as positive, we have

F _{y}= F_{net,y}= F_{y}= n - w - FSince the book sits on a table, we know it does not accelerate so a

_{y}= 0. This meansn - w - F = 0 n = w + F

We can use Newton's Laws to determin the value of the normal force n.

ToCFrictionReturn to ToC, Ch 6, Application of Newton's Laws(c) 2005, Doug Davis; all rights reserved