Newton's 2nd law

Prerequisites

Putting in forces

In our discussion of Inertia we identified two "change equations", one that was relevant when an object was feeling a single unbalanced force for a time interval $Δt$, and one that was relevant when the object was only feeling forces that were all balanced for a time interval $Δt$.

$ΔI/m = Δv$.  (During a time interval when an unbalanced influence is acting)

$Δx = v_0Δt$.   (During a time interval when no unbalanced influence is acting)

where $ΔI$ is an impulse provided to the object by interactions with other objects, and m is the mass of the object. In our discussion Quantifying impulse and force we decided that when there was one unbalanced force acting on an object, that impulse was given by

$$ΔI = FΔt$$

a force times the time interval.  If the time interval is long enough that we can't treat the force as unchanging during the interval, we might want to call this the average force, $\langle F\rangle$. This gives us the two equations

$Δx = v_0Δt$   (During a time interval when no unbalanced force is acting)

$\langle F \rangleΔt /m = Δv$  (During a time interval when a single unbalanced force is acting)

Dividing both sides of each equation by $Δt$ we get

$v_0 = \frac{Δx}{Δt}$.   (During a time interval when no unbalanced force is acting)

$\frac{Δv}{Δt} = \langle F\rangle/m$  (During a time interval when a single unbalanced force is acting)

The first looks like the definition of velocity, but we don't have the "average" marker.  So instead of being a definition, we can interpret this as saying, "when there are no unbalanced forces, the velocity is constant."  The second looks like the definition of acceleration, but it has something else on the right.  As a result, we can interpret this as saying, "an unbalanced force causes an acceleration: $\langle a\rangle = \langle F\rangle /m$".

Going instantaneous

If we let our time intervals get very small, we can drop the averages and replace our deltas by derivatives.  The equations become

$$v=\frac{dx}{dt}\;\;\;\;\;a=\frac{F}{m}$$

Notice that the meanings of these changes are different.  The first has just become the definition of velocity in a tiny interval.  The second, however, is something else. This is NOT a definition, but rather a physical law: we are stating not what is the definition of acceleration (which is $a = dv/dt$) but rather what CAUSES acceleration.

Motion in all directions: a vector law

When we set this up we restricted our considerations to motion along a single line.  What happens when we lift that restriction? The key idea is fairly natural:

The force acting in a given direction only acts to change the motion in that direction.

This becomes particularly useful when we use vector coordinates, breaking things up into motion in an x-direction and a y-direction:

The force acting in the x-direction only acts to change the motion in the x-direction;

the force acting in the y-direction only acts to change the motion in the y-direction.

(and of course if we described motion in a third dimension we would have the z-direction statement as well).  If we put these into equations these become

$$v_x=\frac{dx}{dt}\;\;\;\;\;a_x=\frac{F_x}{m}$$

$$v_y=\frac{dy}{dt}\;\;\;\;\;a_y=\frac{F_y}{m}$$

If we now introduce our vectors for each of position, velocity, acceleration, and force, like this

$$\overrightarrow{r} = x\hat{i} + y\hat{j}\;\;\;\;\;\overrightarrow{v} = v_x\hat{i} + v_y\hat{j}\;\;\;\;\;\overrightarrow{a} = a_x\hat{i} + a_y\hat{j}\;\;\;\;\;\overrightarrow{F} = F_x\hat{i} + F_y\hat{j}$$

we can summarize our four equations (two each about x and y) as a pair of vector equations:

$$\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}\;\;\;\;\;\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt} = \frac{\overrightarrow{F}}{m}$$

Multiple forces: the net force

We worked all the above out by looking at what happened when we pulled a cart with a single constant force — a stretched spring.  The next question is: what do we have to do when there are many different forces?  As we have seen in Kinds of Forces, every object typically interacts with lots of other objects and many forces.  What do we do then?  As we saw in our qualitative discussion, Superposition, and in our quantitative discussion, Adding forces, forces have magnitude and direction, so a natural mathematical model to apply is to treat them as vectors.  The result is that when we have many forces acting on a particular object (call it object A), the force we want is the vector sum of all the forces acting on A. (Remember object egotism!) We call this vector sum the net force acting on A.

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

where the sum over j is taken over all the objects that interact with object A. Since the object will respond to the effect of ALL the forces it feels, we have to put in the net force as the factor that causes an object to change its motion.

Newton's Second Law

The combination of all these ideas results in Newton's second law, probably the most important law in the history of physics, not just because it gives us a framework to describe so much of the observable world, but because it set the style for the entire development of physics -- a powerful combination of qualitative and quantitative reasoning.  The resulting law can be written very simply.

Newton's 2nd Law:  When an object A is subjected to a set of forces, if those forces are unbalanced, it responds to them by changing its velocity according to the following equation:

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

Joe Redish, Ben Dreyfus, Julia Gouvea, and Karen Nordstrom 8/21/13

Workout: Newton's 2nd law

 

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