A force is exerted on a box and an equal and opposite force is exerted by the box what will happen

If I could only change one thing about physics education, it would be the phrasing of Newton's 3rd law. According to my copy of Magnificent Principia (by Colin Pask, Prometheus Books, 2013) the "To every action there is always opposed an equal reaction..." phrasing is Newton's. And it's been causing confusion ever since.

To get a sense of what Newton really meant, consider the universal gravitation equation: $$F=G\frac{m_1m_2}{r^2}$$

Notice there are two masses specified, but there is no "source" mass and no "target" mass. And there is only one force produced by this equation. Now, you can look at it as two different forces: $m_1$ attracting $m_2$ and $m_2$ attracting $m_1$. But that is misleading. It gives the impression that the forces somehow have independent existences. But they don't. They are completely, inextricably linked. So much so, that I think it makes much more sense to this of this as one attractive force between two masses.

Coulombs law follows the same format:

$$F=k_c\frac{q_1q_2}{r^2}$$

Again, you can think of this as two different forces. But I think the equation really hints at a single attractive force (different charge signs) or a single repulsive force (identical charge signs) between two charges.

That is what Newton meant by his third law. It's not possible for $m_1$ to attract $m_2$ without $m_1$ being caught up in the very same force of attraction between the two particles. And it's not possible for $q_1$ to attract or repel $q_2$ without $q_1$ being caught up in the very same force.

Newton's third law is traditionally taught as pairs of forces. I think it makes more sense to present it is as a single force that must always operate between pairs of bodies, as implied by Coulomb's law and the Universal Gravitation equation.

This is harder to see with contact forces. Part of the problem is that human muscles must constantly expend energy at a molecular level in order to stay contracted. So it's easy to confuse force exertion with expenditure of energy. And humans have cognition and agency. So to say, "The person pushes on the matchbox and the matchbox pushes on the person" feels wrong because the person is expending energy; the matchbox is not. The person has agency and initiates the push; the matchbox is inanimate.

To get a better feel for Newton's third law, consider yourself in a deep swimming pool where your feet are off the bottom. You're next to the wall. Now push on the wall. What happens? You push yourself away from the wall. The traditional explanation is that you push on the wall, and "the wall pushes back on you." And while that is technically true, it doesn't make intuitive sense because you know darn well that you're the one doing the pushing.

What's really happening is that you create a repulsive force between the wall and yourself. The wall is fixed to the earth and the earth is mighty big and hard to move. So the repulsive force manifests itself in you pushing yourself away from the wall.

When you "push the matchbox," you're really setting up a repulsive force between your finger and the matchbox. (At a molecular level, this is just the Coulomb repulsion, of course.) But you're much more massive than the matchbox. Your weight and the friction between your shoes and the floor essentially fix you to the floor and make you immovable. So the repulsive force manifests itself as the matchbox moving.

Finally, when dealing with forces where one mass (or one charge) is so much larger than the other (such as an apple falling towards the earth) it's very common to ignore that fact that the masses are attracting each other, and to phrase the interaction as if it were just the earth attracting the apple and nothing more. That is an oversimplification. But it's justified by the fact that the attractive force between the two masses is overwhelmingly manifested in the motion of the apple.

In fact, Newton phrased that part well in The Principia,

"The changes made by these actions are equal . . . if the bodies are not hindered by any other impediments . . . the changes of velocities made towards common parts are reciprocally proportional to the bodies [the masses]."

Forces Due to Friction (and Newton's Third Law)

There are two forms of force due to friction, static friction and sliding friction. When you push a heavy box, it pushes back at you with an equal and opposite force (Third Law) so that the harder the force of your action, the greater the force of reaction until you apply a force great enough to cause the box to begin sliding. You then notice that it requires less force to cause the box to continue to slide.

The force of static friction is what pushes your car forward. The engine provides the force to turn the tires which, in turn, pushes backwards against the road surface. By Newton's Third Law, the "reaction" of the surface to the turning wheel is to provide a forward force of equal magnitude to the force of the wheel pushing backwards against the road surface. This is a force of static friction as long as the wheel is not slipping.

The size of the friction force depends on the weight of the object. In the case of static friction, the maximum friction force occurs just before slipping. Its magnitude is the weight of the object times the coefficient of static friction. We will do exercises only for cases with sliding friction. In that case, the force of sliding friction is given by the coefficient of sliding friction times the weight of the object. The coefficients of static and sliding friction depend on the properties of the object's surface, as well as the property of the surface on which it is resting. Clearly, resting on sandpaper would be expected to give a different answer than resting on ice.

For those who are following this closely, consider how anti-lock brakes work. When you apply your car brakes, you want the greatest possible friction force to oppose the car's motion. This occurs when the wheels are in contact with the surface, rather when they are skidding, or sliding. So you want the wheels to keeps spinning and not to lock...i.e., to stop turning at the rate the car is moving forward. With computer controls, anti-lock breaks are designed to keep the wheels rolling while still applying braking force needed to slow down the car.

A Closer Look at Newton's Third Law

The Third Law says that forces come in pairs. When an object A exerts a force on object B, object B exerts an equal and opposite force on object A. For example, when an object is attracted by the earth's gravitational force, the object attracts the earth with an equal an opposite force. According to Newton's second law, an object's weight (W) causes it to accelerate towards the earth at the rate given by g = W/m = 9.8 meters / s2, where m is the object's mass. However, what is not readily realized is that the earth is also accelerating toward the object at a rate given by W/Me, where Me is the earth's mass. Since Me is so incredibly large compared with the mass of an ordinary object, the earth's acceleration toward the object is negligible for all practical considerations.

The Third Law if often stated by saying the for every "action" there is an equal and opposite "reaction."

Falling objects accelerate toward the earth, but what about objects at rest on the earth, what prevents them from moving? The net force must be zero if they don't move, but how is the force of gravity counterbalanced? See Figure 2-16 of page 45 in the text.

The person in the figure is standing at rest on a platform. He experiences a force Wep (earth-on-person) and the earth experiences a force Wpe (person-on-earth). Wep and Wpe are a pair of Third Law forces. They act on different bodies. The earth attracts the person, and the person attracts the earth.

The person also presses against the floor with a force equal to Wep, his weight. We call this force, Fpf (person-on-floor). But now the Third Law enters again. The reaction to this force is Ffp (floor-on-person).

Now consider Newton's Second Law as it applies to the motion of the person. The net force acting on the person is his weight, Wep pointing downward, counterbalanced by the force Ffp of the floor acting upward. The two cancel, so the net force is zero and his acceleration is zero...i.e.,.he remains at rest.

A rocket is propelled in accordance with Newton's Third Law. A force is required to eject the rocket gas, Frg (rocket-on-gas). This is counterbalanced by the force of the gas on the rocket, Fgr (gas-on-rocket). In empty space, Fgr is the net force acting on the rocket and it is accelerated at the rate Ar (acceleration of rocket) where Fgr = Mr x Ar (2nd Law), where Mr is the mass of the rocket.

Another Third Law example is that of a bullet fired out of a rifle. The force exerted by the expanding gas in the rifle on the bullet is equal and opposite to the force exerted by the bullet back on the rifle. The bullet is much less massive than the rifle, and the person holding the rifle, so it accelerates very rapidly. The rifle and the person are also accelerated by the recoil force, but much less so because of their much greater mass.

Newton's Laws for Circular Motion

• Newton's Third Law explains force of moon on earth is equal and opposite to the force of earth on moon. Likewise, force of sun on earth-moon system is equal and opposite to force of earth-moon system on sun.
• Newton's Second Law relates the force directed toward the center of the circle to the acceleration of the rotating object whose direction also points to the center.
• The attactive force acting between an satellite orbiting the earth in a circular path causes the object to accelerate toward the earth, but as it keeps its circular orbit, it never gets closer to the earth! This is because the satellite has a velocity tangent to its circular orbit which tends to cause it to move away from the earth. If gravity were suddenly turned off, like the knife cutting the string in the example discussed in class, the satellite would maintain a straight line motion (Newton's First Law) and travel away from the earth. The gravitational pull keeps the satellite in orbit, causing it to fall by just the right amount in each instant of time. Note that a circular orbit is achieved if the tangential velocity is exactly the right amount for the satellite's distance above the earth. If the tangential velocity were greater than this amount, it would not maintain a circular orbit ... it would form an elliptic orbit if the velocity were not too great. For great enough velocity, the object would move an ever increasing distance away from the earth. Likewise, for a tangential velocity below the critical value, the object would fall increasingly closer to the earth.