“A body remains at rest, or in motion at a constant speed in a straight line, unless it is acted upon by a force.”
Strangely enough, this does not appear to reflect upon the real world. Why is that?
It’s very known that reality is much more complex than scientific theories may presume. In fact, in reality a body may be affected by other inconvenient external forces such as air resistance, gravity, etc. Anyhow, the system within this law’s statement is such that there is no inconvenient forces acting upon the body - it is an mind experiment, as one would call it.
Imagine the perfect scenario where a perfect sphere is currently idle at the top of a perfectly friction-less horizontal plane at complete vacuum. As Newton’s First Law presumes, it shouldn’t move until some force is applied unto it. Now picture a horizontal force (in respect to the floor/plane) being applied to the sphere, now it should start moving as it is gaining acceleration. If the object applying the force suddenly vanishes, then it should move with a constant speed forever (well, at least until another force is applied).
“At any instant of time, the net force on a body is equal to the body’s acceleration multiplied by its mass ($F_{\text{net}} = m \cdot a$) or, equivalently, the rate at which the body’s momentum is changing with time.”
By definition, the momentum $P$ at certain instant of time $t$ of a body is the product of it’s mass $m$ and it’s velocity $v_t$. The law also states that the net force on a body is equal to the rate at which the body’s momentum is changing with time, therefore:
$$
P = m \cdot v_t \implies F = \frac{d P}{d t} = \frac{d(m \cdot v_t)}{d t} = m \cdot \frac{d v_t}{d t} = m \cdot a
$$
Hence, it is now proved Newton’s Second Law of Motion! Or isn’t it? Well, experience tells us this is true for rigid macro-sized bodies with relatively small speed, i.e., it is valid for “everyday situations”.
The applications of this law are vast. Indeed, it is often held as the most important law of the three. Let’s see one application of it:
Problem:
Let $M$ be a mechanical engine whose function is to pull threads, $A$ an body of mass $m_A = 10^3 kg$ and $B$ a counterweight-body going down (with mass $m_B = 5 \cdot 10^2 kg$), such as the diagram below. Assume there is no air resistance, the threads are ideal (i.e., are weightless, flexible and inextensible) and the gravity $\overrightarrow{g}$ has module $10 m/s^2$. Find the module of $F_M$, if $A$ is accelerated upwards (such acceleration has module $a = 0.2 m/s^2$).
Solution:
First we need to find the net force $F_A$ of $A$, which is such that
$$
F_A = F_M + F_T - W_A = m_A \cdot a = 10^3 kg \cdot 0.2 m/s^2 = 200 N
$$
where $F_T$ is the traction force which the thread applies onto $A$ and $W_A = m_A \cdot g = 10^4 kg$ is the weight of $A$. In order to find $F_T$, we need to find the net force $F_B$ of $B$.
$$
F_B = W_B - F_T = m_B \cdot a = ( 5.10^2 kg ) \cdot 0.2 m/s^2 = 100 N
\newline
\implies F_T = W_B - ( m_B \cdot a ) = 5 \cdot 10^3 N - 100N = 4900 N
$$
Now it’s just an algebra play:
$$
F_M = W_A + 200 N - F_T = ( 10^4 + 200 - 4900 ) N = 5300 N
$$
Hence, the module/intensity of $F_M$ is $5300 N$.
“If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.”
Look at the scheme:
If our Actor pushes the box with force $\overrightarrow{F}$, then the box will apply onto the Actor a force $-\overrightarrow{F}$ (i.e., a force with same intensity but opposite direction).
Newton’s Third Law can be restated as “every action produces a correspondent ‘opposite’ reaction”, which implies that forces exist in “pairs”.
You may have noticed the post’s description:
Newton’s Laws of Motion
Does the universe has an even or odd number of Newtonian forces?
April 21, 2025
Here’s an answer: Since forces exist in “pairs”, then at Newton’s POV the universe should have an even number of forces.
Back to our average undergraduate physics study-session, one question wander at the back of our minds: Why is that when one does push-ups the earth isn’t pushed down? Well, it is, actually! Considering Newton’s Second Law, it is easy to see that the acceleration of a body is inversely proportional to it’s mass. Hence, since Planet Earth has a massive amount of mass, then the acceleration provoked by the force of a mere human pushing it is just despicable.
