# Reading the content in Newton's 2nd law

#### Prerequisite

Newton's second law is written quite simply:

$$\overrightarrow{a}_A= \frac{\overrightarrow{F}^{net}_A}{m_A}$$

As well as being a way to calculate motion quantitatively, there is a large set of qualiitative conceptual ideas packed into interpreted and making sense of what this equation is telling us. Each symbol in the equation reminds us of some important conceptual idea. Let's make sense of Newton's 2nd Law — "see the dog in the equation" — by unpacking what you have to know in order to understand this simple-appearing relationship.

1. a -- The thing on the left of the equation is the acceleration.  To understand that, we have to understand the whole array of specifying an object's position (coordinates) and how that position changes (derivativesvelocityacceleration). This means (for motion in one dimension) we need the definitions

$$v=\frac{dx}{dt}\;\;\;\;\;a=\frac{dv}{dt}$$

It's important to note that the acceleration is written on the left.  We do this to remind ourselves that it's the forces that cause the acceleration rather than the other way around.  Though of course if we know the acceleration and mass we can find the net force.

2. A -- Each of the variables has a subscript labeling which object we are talking about.  This reminds us that a fundamental assumption of the Newtonian framework is that we best understand what is happening by considering individual objects and figuring out what influences are acting on them (Object egotism). Each object we consider will have its own Newton 2 equation.  The subscript A on Fnet reminds us that it is the forces that the object feels that control its motion.  (The forces it exerts have effects on the motion of the objects it exerts them on.)

3. F -- To interpret this we need to understand that it is the interactions with other objects that cause the object we are considering to change its motion (accelerate). And we need to understand how this force is quantified by an operational definition.

4. net -- This little superscript holds a lot of conceptual ideas.  First, that it is the (vector) sum of the forces that an object feels that results in its acceleration.  Each individual force does not produce an individual acceleration.  When we break out this sum explicitly,

$$\overrightarrow{F}^{net}_A = \overrightarrow{F}_{B\rightarrow A}+ \overrightarrow{F}_{C\rightarrow A} + \overrightarrow{F}_{D\rightarrow A} + ... = \sum_j \overrightarrow{F}_{j\rightarrow A}$$

the subscripts on the individual forces remind us that every force is caused by another object.  Further, the forces we want to include are all the forces exerted by other objects on the object we are considering.

5. m -- Dividing the net force by m (subscript A) reminds us that the resulting force on the object is shared over the parts of the object.  A bigger object will have less of a response (acceleration) to the same force.

6. --The little arrows on top of the acceleration and net force remind us that Newton's second law is a vector equation.  This means that each perpendicular direction has its own Newton's law -- $x$, $y$, and $z$.  Further, that it is the net force in the $x$ direction that affects the motion in the $x$ direction, the net force in the $y$ direction that affects the motion in the $y$ direction, etc.

$$a_x=\frac{F_x}{m}\;\;\;\;\;a_y=\frac{F_y}{m}\;\;\;\;\;a_z=\frac{F_z}{m}$$

That's a lot to pack into one little equation with what looks like 3 symbols (that turn out to be 6).  But each of these ideas is an essential piece of making sense of this important principle.

When does this equation hold? All of Newton's Laws have extraordinary power: they hold for any object whose internal structure can be ignored. If the object's internal structure cannot be ignored, then these principles hold for its internal parts (as long as their structure can be ignored). If an object has parts, and those parts are moving and interacting the center of mass of the whole object still satisfies Newton's Laws. For molecular motions, you can often ignore the atoms and just look at the molecules. Then Newton's Laws hold for them. If you can't ignore the molecular structure, then you have to treat the atoms as the objects to be considered. Unless they change their bonding structure (and not just the orientation of the bonds), they can still be treated with Newton's Laws. Beyond that you need Quantum Mechanics – and even that does not violate some of the results that follow from Newton's Laws: for example, momentum conservation. Newton's laws also need to be modified when objects start moving at significant fractions of the speed of light (3 x 108 m/s). But for biological situations, Newton's second law can essentially always be trusted.

Joe Redish 9/20/11

Article 361