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2.2: Effects of Force

Newton's Second Law

We have built some tools for analyzing situations where forces act on objects (force phrase, FBDs), and we know that there can only be accelerations when forces are present (first law).  But we still are not yet able to describe the motion of an object under the influence of one or more forces.  That’s because the first law only tells us qualitatively what is happening.  In physics we seek to develop quantitative models, and that’s where the second law comes in.  It is really just a more detailed description of the first law, or alternatively, the first law is just a special case of the second law.

We know that force is related not to velocity (because the first law says that constant velocity exists in the absence of force), but rather the change of velocity. More specifically, the rate of change of the velocity – the acceleration. Newton defined force in the simplest possible fashion in terms of acceleration – with a linear relationship.  He knew that pushing equal amounts on two objects of different masses resulted in different changes of motion, so he stated that the relationship between force and acceleration as a simple proportionality:

$acceleration \ of \ object \ = \frac{force \ acting \ on \ object}{mass \ of \ object}$

The idea is that for a given force, the reaction of the object (in the form of an acceleration) is inversely-proportional to the amount of mass the object possesses.  Let’s take a moment to mention units:

$$\left[ F \right] = \dfrac{kg \cdot m}{s^2} = \; "Newtons" \; \left( N \right)$$

There is much more detail lurking in here.  First of all, acceleration and force are both vectors, while mass is a scalar, so the second law is actually a vector equation:

$\overrightarrow a = \frac{\overrightarrow F}{m}$

This means that the acceleration experienced by an object is just a scaled vector of the force exerted on the object. That is, the acceleration and the force always point in the same direction (mass is never negative).  Of course, the scaling also changes the units.

Most people first encounter Newton's second law expressed as $$\overrightarrow F = m \overrightarrow a$$.  While this is mathematically equivalent to what is above, it is very dangerous to write this way, as it encourages a very common misconception.  We write it as we do above to emphasize the interpretation: "the effect on the motion (the acceleration) results from the cause (the force), moderated by a property of the object experiencing the effect (the mass)."  The danger of using the other expression is that it reads like, "the force of the object equals the mass of the object multiplied by the acceleration of the object."  This turns the quantities of force and acceleration into properties of the object, rather than cause and effect, and this leads to subtle but important misconceptions.

We aren’t done modifying the second law to its proper form yet!  We know from our discussion of force diagrams that a large number of forces can be on an object at the same time.  Which force is the one that causes the acceleration?  All of them.  Do we figure out the accelerations of each force and then add them up?  That makes no sense physically – objects do not experience lots of accelerations at once.  Instead, we take all of the forces together and add them as vectors to create a single composite force we call the net force, and that is what goes into the equation:

$\overrightarrow a = \dfrac{\overrightarrow F_{net}}{m}$

Still not done!  Suppose only a single force is acting on a specific point of an extended object?  Remarkably, it doesn’t matter where the force acts on an object, if you measure the motion of the center of mass of the object (a quantity we will examine in more detail later on), it is the acceleration of that point that is determined by the force and mass.

Figure 2.2.1 – What Part of an Extended Object Accelerates According to the Second Law?

So finally we have arrived at the full-blown version of Newton's second law of motion:

$\overrightarrow a_{cm} = \dfrac{\overrightarrow F_{net}}{m}$

Now perhaps the main point of the free-body diagram is clear: The diagram facilitates our ability to add force vectors (all of which are on the object, not by it), giving us the net force acting on it.  We use the usual tools for vector addition to obtain this net force, perhaps expressing it in terms of its components in some coordinate system.  Then we divide this vector by the mass of the object, and we have the acceleration of its center of mass, which could also be expressed in terms of its components.  Then the acceleration is used to describe the object's motion as we have used it in kinematics.

Example $$\PageIndex{1}$$

A driver steps on the brake pedal of her car, slowing the car down, and her body experiences an acceleration as a result.  Which of the following forces does Newton’s 2nd Law include when determining her acceleration?

1. normal force by driver’s foot on the brake pedal
2. friction force by the car tires on the road
3. friction force by the road on the car tires
4. all of these
5. none of these
Solution

(e)  One must be very precise when it comes to identifying forces, as ultimately they must be plugged into a mathematical formula.  It is not enough that a force sets off a chain of events that leads to an acceleration, it must be the direct effect of that acceleration in order to be the force that is featured in the second law.  In addition to being a direct force, it isn't even sufficient to isolate the correct interaction – the specific "twin" from the third law force pair must be identified.  That is, the force must be on the object in order to accelerate it.  The problem asks what force slows down her body.  The normal force on the brake pedal affects the motion of the brake pedal.  The friction force on the road affects the motion of the road.  The friction force on the tires (which are part of the car's system) affects the motion of the car system.  While parts of the car system (namely tension by the seatbelt, friction by the car seat, and normal force by the steering wheel do slow her down, and the friction force on the tires slows down the car, this chain of events does not mean that the friction force slows her down.  If you plug the friction force on the tires and her body's mass into the second law, the acceleration you calculate for her will not be correct.

Example $$\PageIndex{2}$$

A boy throws a ball straight up, and catches it when it returns.  Which pair of diagrams best represents the directions of the net forces experienced by the ball when it hits the peak of its flight (i.e. when it isn’t moving), and while the boy is catching it (i.e. not after he has caught it)?

Solution

(d)  Gravity is always acting on the ball, no matter where it is.  When it reaches its peak, there are no other forces on it (the boy’s hand is no longer in contact with it), so the gravity force is the net force, and it points down.  When the ball is in the process of being caught, there is a force up on it by the boy’s hand, and since the ball is moving downward and is slowing, it is accelerating upward, which means the force from the boy's hand exceeds the force of gravity and the net force is upward.

Combining Newton's Second and Third Laws

We already saw in the case of the tug-o-war how the second and third laws work together: Each participant exerts an equal tension force on the other. These forces don't cancel because according to the second law, it is only the forces on the system that cause it to accelerate, and these two forces act on different systems.  One individual feels this tension force on them, and if it is the only force, it will accelerate them forward (i.e. they will lose).  To counter this, this person pushes the ground forward with their feet.  But this is a force on the ground, which can only affect the acceleration of the ground, not the person, according to the second law.  But now the third law comes to the rescue – when the person pushes on the ground, the ground pushes back equally, and this force is on the person.  This opposes the pull of the rope, and if it is stronger than the rope's pull, the net force on the person is backward, and that person wins.

Another question that can be very confusing at first about the third law is this:

Another Puzzling Question: If the gravity force on a rock by the earth is equal and opposite to the gravity force on the earth by the rock, then why doesn't the earth accelerate upward when a rock is dropped to the ground?

As you might have guessed from the title of this section, it is the second law that comes to the rescue here.  The answer is that the earth does accelerate up toward the rock!  It feels the same net force that the rock feels, thanks to the third law.  But when it comes to the acceleration, we need to divide this net force by the mass of the accelerated object, and since the mass of the earth is so much greater than that of the rock, it experiences a much, much smaller acceleration – so much smaller that it is imperceptible.