Newton's Laws of Motion

Perhaps the most heavily used equation in physics is Newton's second law of motion:

$\displaystyle \zbox {\mbox{\emph{Force = Mass $\times$\ Acceleration}}}
$

That is, when a force is applied to a mass, the mass experiences an acceleration proportional to the applied force. Denoting the mass by $ m$, force at time $ t$ by $ f(t)$, and acceleration by

$\displaystyle a(t)\isdefs {\ddot x}(t) \isdefs \frac{d^2 x(t)}{dt^2},
$

we have

$\displaystyle \zbox {f(t) = m\,a(t) = m\,{\ddot x}(t).} \protect$ (B.1)

In this formulation, the applied force $ f(t)$ is considered positive in the direction of positive mass-position $ x(t)$. The force $ f(t)$ and acceleration $ a(t)$ are, in general, vectors in three-dimensional space $ x\in{\bf R}^3$. In other words, force and acceleration are generally vector-valued functions of time $ t$. The mass $ m$ is a scalar quantity, and can be considered a measure of the inertia of the physical system (see §B.1.1 below).

Newton's Three Laws of Motion

Newton's three laws of motion may be stated as follows:

  1. Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it.

  2. Force equals mass times acceleration [ $ f(t)=m\,a(t)$].

  3. For every action there is an equal and opposite reaction.

The first law, also called the law of inertia, was pioneered by Galileo. This was quite a conceptual leap because it was not possible in Galileo's time to observe a moving object without at least some frictional forces dragging against the motion. In fact, for over a thousand years before Galileo, educated individuals believed Aristotle's formulation that, wherever there is motion, there is an external force producing that motion.

The second law, $ f(t)=m\,a(t)$, actually implies the first law, since when $ f(t)=0$ (no applied force), the acceleration $ a(t)$ is zero, implying a constant velocity $ v(t)$. (The velocity is simply the integral with respect to time of $ a(t)={\dot v}(t)$.)

Newton's third law implies conservation of momentum [137]. It can also be seen as following from the second law: When one object ``pushes'' a second object at some (massless) point of contact using an applied force, there must be an equal and opposite force from the second object that cancels the applied force. Otherwise, there would be a nonzero net force on a massless point which, by the second law, would accelerate the point of contact by an infinite amount.

In summary, Newton's laws boil down to $ f=ma$. An enormous quantity of physical science has been developed by applying this simpleB.1 mathematical law to different physical situations.


Mass

Mass is an intrinsic property of matter. From Newton's second law, $ f(t)=m\,a(t)$, we have that the amount of force required to accelerate an object, by a given amount, is proportional to its mass. Thus, the mass of an object quantifies its inertia--its resistance to a change in velocity.

We can measure the mass of an object by measuring the gravitational force between it and another known mass, as described in the next section. This is a special case of measuring its acceleration in response to a known force. Whatever the force $ f$, the mass $ m$ is given by $ f$ divided by the resulting acceleration $ a$, again by Newton's second law $ f=ma$.

The usual mathematical model for an ideal mass is a dimensionless point at some location in space. While no real objects are dimensionless, they can often be treated mathematically as dimensionless points located at their center of mass, or centroidB.4.1).

The physical state of a mass $ m$ at time $ t$ consists of its position $ x(t)$ and velocity $ {\dot x}(t)$ in 3D space. The amount of mass itself, $ m$, is regarded as a fixed parameter that does not change. In other words, the state $ (x,{\dot x})$ of a physical system typically changes over time, while any parameters of the system, such as mass $ m$, remain fixed over time (unless otherwise specified).


Gravitational Force

We are all familiar with the force of gravity. It is a fundamental observed property of our universe that any two masses $ m_1$ and $ m_2$ experience an attracting force $ f$ given by the formula

$\displaystyle f(t) = G\frac{m_1 m_2}{r^2(t)} \protect$ (B.2)

where $ r(t)$ is the distance between the centroids of the masses $ m_1$ and $ m_2$ at time $ t$, and $ G$ is the gravitation constant.B.2

The law of gravitation Eq.$ \,$(B.2) can be accepted as an experimental fact which defines the concept of a force.B.3 The giant conceptual leap taken by Newton was that the law of gravitation is universal--applying to celestial bodies as well as objects on earth. When a mass is ``dropped'' and allowed to ``fall'' in a gravitational field, it is observed to experience a uniform acceleration proportional to its mass. Newton's second law of motion (§B.1) quantifies this result.


Hooke's Law

Consider an ideal spring suspending a mass from a rigid ceiling, as depicted in Fig.B.1. Assume the mass is at rest, and that its distance $ x(t)$ from the ceiling is fixed.

Figure B.1: Mass hung by a spring from a rigid ceiling.
\includegraphics{eps/springmass-gravity}

If $ M$ denotes the mass of the earth, and $ r$ is the distance of mass $ m$'s center from the earth's center of mass, then the downward force on the mass $ m$ is given by Eq.$ \,$(B.2) as

$\displaystyle f_m = g m,
$

where

$\displaystyle g = G\frac{M}{r^2}
$

is called the acceleration due to gravity. Changes in $ r$ due to the motion of the mass are assumed negligible relative to the radius of the earth (about $ 4000$ miles), and so $ g$ is treated as a constant for most practical purposes near the earth's surface. We see that if we double the mass $ m$, we double the force $ f_m$ pulling on the spring. It is an experimental fact that typical springs exhibit a displacement $ x_m$ that is approximately proportional to the applied force $ f_m$ for a wide range of applied forces. This is Hooke's law for ideal springs:

$\displaystyle \zbox {f(t) = k\,x_m(t),}$   (Hooke's Law)$\displaystyle \protect$ (B.3)

where $ x_m(t)$ is the displacement of the spring from its natural length. We call $ k$ the spring constant, or stiffness of the spring. In terms of our previous notation, we have

$\displaystyle x(t) = x_k + x_m(t),
$

where $ x_k$ is the length of the spring with no mass attached.

Note that the force on the spring in Fig.B.1 is gravitational force. Equal and opposite to the force of gravity is the spring force exerted upward by the spring on the mass (which is not moving). We know that the spring force is equal and opposite to the gravitational force because the mass would otherwise be accelerated by the net force.B.4 Therefore, like gravity, a displaced spring can be regarded as a definition of an applied force. That is, whenever you have to think of an applied force, you can always consider it as being delivered by the end of some ideal spring attached to some external physical system.

Note, by the way, that normal interaction forces when objects touch arise from the Coulomb force (electrostatic force, or repulsion of like charges) between electron orbitals. This electrostatic force obeys an ``inverse square law'' like gravity, and therefore also behaves like an ideal spring for small displacements.B.5

The specific value of $ k$ depends on the physical units adopted as well as the ``stiffness'' of the spring. What is most important in this definition of force is that a doubling of spring displacement doubles the force. That is, the spring force is a linear function of spring displacement (compression or stretching).


Applying Newton's Laws of Motion

Figure B.2: Mass-spring system.
\includegraphics{eps/springmass-phy}

As a simple example, consider a mass $ m$ driven along a frictionless surface by an ideal spring $ k$, as shown in Fig.B.2. Assume that the mass position $ x=0$ corresponds to the spring at rest, i.e., not stretched or compressed. The force necessary to compress the spring by a distance $ x$ is given by Hooke's lawB.1.3):

$\displaystyle f_k(t) = -k\,x(t)
$

This force is balanced at all times by the inertial force $ f_m(x)=-m{\ddot x}$ of the mass $ m$, i.e. $ f_k+f_m=0$, yieldingB.6

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0\, \quad \forall t\ge 0, \quad x(0)=A, \quad {\dot x}(0)=0, \protect$ (B.4)

where we have defined $ A$ as the initial displacement of the mass along $ x$. This is a differential equation whose solution gives the equation of motion of the mass-spring junction for all time:B.7

$\displaystyle x(t) = A\cos(\omega_0 t), \quad \forall t\ge 0, \protect$ (B.5)

where $ \omega_0\isdeftext \sqrt{k/m}$ denotes the frequency of oscillation in radians per second. More generally, the complete space of solutions to Eq.$ \,$(B.4), corresponding to all possible initial displacements $ x(0)$ and initial velocities $ {\dot x}(0)$, is the set of all sinusoidal oscillations at frequency $ \omega_0$:

$\displaystyle x(t) = A\cos(\omega_0 t + \phi), \quad \forall A,\phi\in{\bf R}.
$

The amplitude of oscillation $ A$ and phase offset $ \phi$ are determined by the initial conditions, i.e., the initial position $ x(0)$ and initial velocity $ {\dot x}(0)$ of the mass (its initial state) when we ``let it go'' or ``push it off'' at time $ t=0$.


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