General Introduction to Applications

Now that we know Newton's Laws of Motion, how do we apply them? How can they let us predict the motion of an object if we know all the forces acting upon it? How can they let us predict the 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 = m a

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Example: Consider a crate being pulled along a frictionless floor (while such a floor is very hard to find, this will still help us understand the concept and we can return to this situation later, after considering friction, and solve it more realistically).

Consider a crate being pulled along a horizontal, frictionless floor. A rope is tied around it and a man pulls on the rope with a force of T. T is the tension in the rope. What happens to the crate?

Before we can apply Newton's Second Law,

F = m a

we must find the net force -- the vector sum of all the forces -- acting on the object. In addition to the force T exerted by the rope, what other forces act on the object?

A good, clear diagram, showing all the forces is an important beginning for every Mechanics 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 pulls down on this crate with a force equal to its weight, w. But the floor supports the crate. The floor responds by pushing up on the crate with a force we call the normal force. "Normal" means "perpendicular". We will call this force n; you may also encounter it labeled N or F_N.

These forces are shown on the "free body diagram" above. We have drawn in all the forces acting on the object. The net force is the vector sum of these forces.

F_net = F = T + n + w

where , the Greek upper case "sigma", means "the sum of". Remember, tho', vector notation is always elegant shorthand notation. When we write

F_net = F = T + n + w

we have really written

F_{net, x} = F_x = T_x + n _x + w _x

and

F_net,y = F _y = T _y + n _y + w _y

What are these x- and y-components of the forces T, n, and w? For this first, simple example, we can find -- by inspection -- that these components are

T_x = T

T _y = 0

n_x = 0

n _y = n

w_x = 0

w _y = - w

where the minus sign means "down".

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

F = m a

But 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 / m

The crate has a horizontal acceleration equal 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, so

n = 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 not always be the case but it is true in this example or in this situation.

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Example: What are the forces acting on book if you push down on it with a force F while 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 -- labeled n in the diagram -- and the force F exerted by the hand. Gravity exerts a force downward, called the weight and labeled w. Just as in the previous example, we can immediately write

F = m a

but that is really elegant shorthand for

F_x = F_net,x = F_x = m a_x

and

F_y = F_net,y = F_y = m a_y

In this example, tho', nothing happens in the horizontal direction. All of the forces have only vertical components so all we really have is

F_y = F_net,y = F_y = m a_y

Taking up as positive, we have

F_y = F_net,y = F_y = n - w - F

Since the book sits on a table, we know it does not accelerate so a_y = 0. This means

n - w - F = 0

n = w + F

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

ToC			Friction
Return to ToC, Ch 6, Application of Newton's Laws

(c) 2005, Doug Davis; all rights reserved