Chapter 5

Newton and 'The Force"

Misconceptions by Aristoteles and followers:

1.

heavier objects fall faster (we have seen how Galileo took care of that)

2.

to keep an object in motion at constant velocity, a constant "agent" is needed (e.g. horse pulling a cart, wind pushing a sailboat,etc.)

The problem with this second statement was in not realizing that, in addition to the pulling or pushing agent, there are other agents, typically friction and air or water resistance, involved in the phenomena mentioned.

If we eliminate such spurious agents, then, once a body is in motion with a certain velocity, it tends to maintain it: in ideal, albeit abstract, conditions, it will move forever at the same velocity. Generalizing this idea, we can state Newton's first law of motion or Law of Inertia :

an object will maintain a state of rest or of motion at constant velocity unless some external agent intervenes to modify it.

Any such agent that can modify the state of motion of an object is what we call a Force.
With hindsight, we can quickly realize that there are plenty of examples of the law of inertia in everyday's life :

• if you (very unwisely) sit in a car without seat belts, and the driver jams the brakes, you fly against the windshield. Why? Is anyone throwing you against the shield? No, what is happening is that, while the car is slowing down, you keep moving with whatever velocity you (and the car) had
• when a car circling a bend encounters an icy patch or an oil slick, it takes off along the tangent to the curve (i.e. in the direction of its instantaneous velocity), since the force that kept it on the curved trajectory, i.e. the tire friction on the ground, stops acting.
• etc.

We then see that a force is something that causes the velocity to change. By definition, whenever an object changes its velocity, it undergoes an acceleration. We then infer that there has to be a strict correlation between forces and accelerations.

Can we say more about such a relation? The first obvious statement is that, the greater the force, the greater the acceleration. Careful measurements would show that the acceleration is actually proportional to the force. Moreover, it is equally intuitive that the acceleration (i.e. the change in velocity) will be in the same direction as the force. We can then write

$\vec{F}\propto\vec{a}$

What else can we say ? We can try applying the same force to different objects, and see what happens . Obviously a more massive object will be accelerated less than a less massive one: try to give the same kick to a soccer ball or to a bowling ball.....Or try to stop (i.e. to change the velocity of) a bowling ball or a soccer ball moving at the same speed...

These considerations allow us to give a quantitative definition to the concept of mass : the mass of an object is a measure of its inertia, i.e of the resistance a body opposes to have its state of motion changed. Mass is therefore the "missing link" in the force vs. acceleration equation, that we can now write

$\vec{F}= m\vec{a}$

This is Newton's second law of motion, and, in spite of its apparent simplicity and almost obvious appearance, it has incredibly far-reaching consequences.

Newton's law allows us to define the unit for forces, the newton (N): a force of 1 newton will impart to the mass of 1 kg the acceleration of 1 m/s². Dimensionally:

[F] = [MLT^-2]

Combination of Forces

In general, there can be many different forces applied to a given body and, in order to obtain the final effect (i.e. the value of the acceleration) one has to consider the net force, i.e. the (vector) sum of all the forces acting onto the body. Most obvious example : if I apply two forces which are equal and opposite, their sum will be zero and no net force will be acting on the body. Does this mean that the object is at rest?

No, if an object has no net force acting upon it, it will be either at rest or moving with constant velocity.

We can then conclude :

no net force $\longrightarrow$ constant velocity (possibly =0)

constant velocity (including 0) $\longrightarrow$ no net force

net force $\neq 0 ~~~\leftrightarrow ~~~$ change in velocity

Problem : stronger, more obedient Malamute and weaker, more unruly Husky pulling a sleigh. Malamute pulls at 10⁰, Husky at 20⁰, but sleighs accelerates dead ahead. What is the ratio of dogs strenghts?
Mass vs. Weight

At this point, it should be clear what we mean by mass. What about weight? We define weight as the force felt by an object of mass m due to gravity. Therefore

$\vec{w} = m\vec{g}$

This definition has the following implications :

• weights should be measured in Newtons (or in pounds, as long as we understand that the pound is a unit of force, not of mass)
• Given that g depends upon the location, the weight of an object is not a costant quantity (while its mass is: mass is an intrinsic, invariant attribute of any given object).
• the correct way to measure a mass is to give it a known force and measure its acceleration.
• astronauts revolving the earth are not weightless (when trying to be precise, one talks in terms of apparent weightlessness), since gravity, albeit reduced, is still very much present. Astronauts are "weightless" as much as you would be if you were standing on bathroom scales in an elevator, and the elevator cable broke: the scales would record 0, but you still are very much under the effect of gravity (and your weight is still mg...)
• when you exercise heavily and burn off some fat, it is more correct to say that you are losing mass. But of course, in any given location, if your mass changes, so does your weight.

Newton's third law

Think of pushing on a wall : you are definitely exerting a force on the wall, but you also feel a pressure on your hand; the wall is therefore exerting a force on your hand. Similarly, if you kick a ball (think of kicking a bowling ball with your bare foot), you will feel for sure that, while you are exerting a force on the ball, the ball gets back at you. These are all examples of a very general rule, Newton's third Law (also called action and reaction), that states :

whenever an object A exerts a force on an object B,
B will exert a force on A
which is equal and opposite to the force exerted by A

Applying Newton's third law can provide the answer to several questions:

• when an apple falls to the ground, what does the earth do?
• how can rocket propulsion work in outer space, where there is no atmosphere for the rocket to push itself against? It is a fact that, as recently as a few decades ago, people (rocket scientists?) claimed that outer space rocket propulsion was impossible since out there the rocket has nothing to push itself against.
• applicaton of the third law sometimes can be tricky:
  suppose I try to move a cart by pushing on it, the cart is going to push on me with equal and opposite force, and we will not go anywhere .....

To solve the quandary, it is essential to realize that we must take into consideration all the forces that are at work. In our case the cart exerts a force on the person; this force is transmitted from the arms to the feet (by the muscle and bone structure) so that the feet push backwards on the earth. But then, by Newton's third law, the earth pushes forward on the feet : if this forward force is larger than the one needed to move the cart, then the cart will move.

When solving problems dealing with Newton's third law you must keep in mind the following rules :

1.

when studying the motion of a body, do take into account all the forces acting on it, BUT

2.

do not confuse between forces that are applied to the body and forces that the body applies to others : the two forces that are mentioned in Newton's third law are not applied to the same body

Normal Force

Look at any object resting on a table : nobody will argue that the object is at rest. Nor anyone will argue that the object is feeling the force of gravity. But, given that the object is at rest, there must be another force acting on the body that cancels the effect of gravity. This force is in fact there, and it is provided by the table itself (if we were to examine the problem at the atomic level, we would realize that this force is due to electrostatic repulsion between the electrons in the object and the table). We conclude that the table produces a force balancing the force that the object is exerting on it. And this is exactly what we expect from Newton's third law : if a body A exerts a force (in this case the weight) on an object B, B will exert an equal and opposite force on A. This force is usually called the normal force (in mathematical jargon normal = perpendicular), since it always acts perpendicularly to the surface generating it.

Important example : given an object of weight w, what is the value of the normal force when the object is resting on a non-horizontal surface?

Equally important is the case when the object and the supporting surface are moving with some acceleration. You all have experienced the feeling of being heavier or lighter when an elevator starts or stops. To examine the problem quantitatively, let us suppose that we ride an elevator while standing on a bathroom scale (the reading on the scale represents the force that you exert on the scale, therefore, by Newton's 3rd law, the force that the scale exerts on you, i.e. the normal force). The question is :

what will the reading of the scale be when the elevator is

1.

at rest ?

2.

moving upward with constant acceleration ?

3.

moving downward with constant acceleration less than g ?

4.

moving down with acceleration g? (the cable has broken !!)

5.

moving up or down with constant velocity ?

The correctness of intuitive answers can be verified by Newton's 2nd law : acceleration is determined by the sum of al the forces acting on you. In this case, the forces are the weight and the normal force, so that we can write (choosing the positive axis to point up)

F_N - mg = ma

F_N = mg + ma = w + ma

Assigning to a all the possible values discussed above we get exactly the results we would expect

Another "hidden" force : Tension

In a large class of problems, we have to deal with objects that are either moved or restrained by means of a rope, a cable, or the like. For all such cases, it is important to realize that the rope itself is at the same time subjected to a force and exerting one (as we should expect anyway, because of Newton's third law).

Example : if you pull a heavy load by means of a rope, you apply a force to the rope (which also means that the rope is applying an equal and opposite force to you). But the force you apply to the rope is transmitted to the load; as the rope applies this force to the load, the load applies the same force onto the rope.... To summarize, there are 4 forces in this problem but (make sure you remark the distinction) two of them are acting onto the rope, while the other two are applied from the rope to external objects. The (equal and opposite) forces applied to and from the rope is what we call the tension.

Examples: what is the rope's tension when

1.

a 50 kg load is hanging from the ceiling

2.

I pull the same load horizonatally, on a frictionless surface, moving with constant velocity

3.

I pull the same load horizonatally, on a frictionless surface, moving with constant acceleration a

More examples :
three weights and two pulleys (static case)
Atwood machine (dynamic case)

Friction

The normal force generated by a surface has the property of not being defined a priori, but to vary according to the force that is applied to the surface (up to a certain limit, after which the surface collapses). Another type of force that has the same property of not being defined a priori, but to respond according to the counterforce applied is the force of friction. Friction plays a fundamental role in everyday's life. On the one side you might think that friction is a nuisance: if there was no friction, you could send loads of arbitrary weight to any far destination while expending hardly any effort : just give it a small nudge in the right direction, and the load will keep moving in that direction forever.... But things are not as simple as that, if there was no friction you would not be able to walk... Or if you wanted to give a nudge to a load to send it somewhere, the load would send you in the opposite direction with equal force....

Resigning ourselves to the existence of friction, we can study its properties:

• when a body is pulled in a certain direction, friction opposes the force by "pulling" in the opposite direction : friction (a force, therefore a vector) acts in the direction opposite to the motion (or attempted motion).
• increasing the pulling force, the frictional force increases accordingly, until one reaches the "breaking point" . We can then say that friction grows in concert with the force applied, until it reaches a maximum value. For a given object, this maximum value depends on the properties of the two surfaces in contact. We quantify this by introducing a coefficient of friction $\mu$: given two bodies pressing on each other, the larger is $\mu$ the larger is the force needed to make them slide across each other.
• experience also tells us that, for a given choice of surfaces in contact, friction will increase proportionally to the strength of the force pressing the bodies together. In other words, friction is proportional to the normal force.

We can summarize all this by :

$F_{friction} \leq \mu F_{normal}$

From this we see that $\mu$ is dimensionless.

Important Remark: the expression above is an equality between vector magnitudes only, and it does not imply equality of direction. In fact friction and normal forces are orthogonal to each other.

Another lesson from experience is that, in general, it takes a bigger effort to put an object into motion than to keep it moving. We account for this effect by referring to static ($\mu_{s}$)and kinetic ($\mu_{k}$) friction coefficients.

Example 1 : a 50 kg crate on an inclined plane starts sliding when the angle is 45 degrees. What is the coefficient of (static) friction ?

Example 2 : when at an angle of 60 degrees, the crate moves with an acceleration g/4. What is the coefficient of (kinetic) friction ?