Newton’s Laws
2.1 Forces and Quantity of Matter
\[ \begin{aligned} \vec{F} = m\vec{a}. \end{aligned} \]
There are four fundamental forces.
| Interaction | Current theory | Mediators | Relative strength | Long-distance behavior (Force/Field) | Range(m) |
|---|---|---|---|---|---|
| Strong | Quantum chromodynamics(QCD) | gluons | \(10^{38}\) | \(\sim r\) (color confinement) | \(10^{-15}\) |
| Electromagnetic | Quantum electrodynamics(QED) | photons | \(10^{36}\) | \(\frac{1}{r^2}\) | \(\infty\) |
| Weak | Electroweak theory(EWT) | W and Z bosons | \(10^{33}\) | \(\frac{1}{r}e^{-m_{W,Z}r}\) | \(10^{-18}\) |
| Gravitation | General relativity(GR) | gravitons(hypothetical) | \(1\) | \(\frac{1}{r^2}\) | \(\infty\) |
\[ \begin{aligned} F_x = -k\Delta x = -k(x - x_{eq}). \end{aligned} \]
Newton’s Law of Universal Gravitation describes the gravitational force between two objects with masses. The force exerted by mass \(m_1\) on mass \(m_2\) due to gravity is described by: \[ \begin{aligned} \vec{F}_{1,2}^G = -G\frac{m_1 m_2}{r_{1,2}^3}\vec{r}_{1,2}. \end{aligned} \]
Close to Earth’s surface, the gravitational attraction between an object and Earth manifests as a downward force whose magnitude is expressed as: \[ \begin{aligned} |F_{\text{earth, object}}^G| = mg, \end{aligned} \]
where \(m\) denotes the mass of the object, and \(g\) represents the acceleration due to Earth’s gravity.
Coulomb’s Law quantifies the electric force between two charged objects. For objects 1 and 2, the force on object 2 due to object 1 is given by: \[ \begin{aligned} \vec{F}_{1,2}^E &= k_e \frac{q_1 q_2}{r_{1,2}^3} \vec{r}_{1,2}. \end{aligned} \] where \(q_1\) and \(q_2\) are the charges, \(\vec{r}_{1,2}\) is the displacement vector pointing from object 1 to object 2, and \(k_e = 8.9875517923(14) \times 10^9 \mathrm{~N \cdot m^2 \cdot C^{-2}}\) is Coulomb’s constant.
\[ \begin{aligned} f_k &= \mu_k N, \end{aligned} \]
\[ \begin{aligned} 0 \le f_s &\le (f_s)_{\max} = \mu_s N. \end{aligned} \]
2.2 Newton’s Laws
An object remains in a state of rest or in uniform motion in a straight line unless it is compelled to change that state by forces acted upon it.
A reference frame in which the first law is valid is called an inertial or Galilean system.
The change of motion is proportional to the motive force impressed and is made in the direction of the right line in which that force is applied.
Namely, we have
\[ \begin{aligned} \vec{F} &= \frac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt} = m\vec{a} \end{aligned} \]
where we assume that the mass \(m\) does not change.
To every action, there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal in magnitude and directed to contrary direction on the right line connecting the two bodies.
2.3 Non-Inertial Frames and Effective Forces
Consider a non-inertial reference frame \(S'\) that translates relative to an inertial frame \(S\) with an acceleration \(\vec{a}_0\). The transformation of acceleration between these frames is given by
\[ \begin{aligned} \vec{a} = \vec{a}' + \vec{a}_0. \end{aligned} \]
Applying Newton’s Second Law in the inertial frame \(S\), we obtain
\[ \begin{aligned} \vec{F} = m\vec{a} = m(\vec{a}' + \vec{a}_0). \end{aligned} \]
Rearranging this equation, we can express the equation of motion in the non-inertial frame \(S'\) as
\[ \begin{aligned} m\vec{a}' = \vec{F} + \vec{F}_a, \end{aligned} \]
where
\[ \begin{aligned} \vec{F}_a = -m\vec{a}_0 \end{aligned} \]
is the effective force, or fictitious force, in the non-inertial frame \(S'\).
- Coordinate transformations:
\[ \begin{aligned} r = r', \quad \dot{r} = \dot{r}', \quad \ddot{r} = \ddot{r}', \\ \theta = \theta' + \omega t, \quad \dot{\theta} = \dot{\theta}' + \omega, \quad \ddot{\theta} = \ddot{\theta}'. \end{aligned} \]
- Unit vector transformations:
\[ \begin{aligned} \hat{r} = \hat{r}', \\ \hat{\theta} = \hat{\theta}'. \end{aligned} \]
In the inertial frame \(S\), the position vector is \(\vec{r} = r \hat{r}\). Taking the time derivative in the inertial frame:
\[ \begin{aligned} \vec{v} = \frac{d\vec{r}}{dt} = \dot{r} \hat{r} + r \dot{\hat{r}}. \end{aligned} \]
Using the relation for the derivative of a unit vector in polar coordinates, \(\dot{\hat{r}} = \dot{\theta} \hat{\theta}\). Substituting \(\dot{\theta} = \dot{\theta}' + \omega\):
\[ \begin{aligned} \vec{v} &= \dot{r} \hat{r} + r(\dot{\theta}' + \omega) \hat{\theta} = (\dot{r} \hat{r} + r \dot{\theta}' \hat{\theta}) + r \omega \hat{\theta}. \end{aligned} \]
Since \(\vec{v}' = \dot{r}' \hat{r}' + r' \dot{\theta}' \hat{\theta}'\) and \(\vec{\omega} \times \vec{r} = r \omega \hat{\theta}\), we obtain:
\[ \begin{aligned} \vec{v} = \vec{v}' + \vec{\omega} \times \vec{r}. \end{aligned} \]
Then \[ \begin{aligned} \vec{a}_r &= (\ddot{r} - r\dot{\theta}^2)\hat{r} = [\ddot{r}' - r'(\dot{\theta}' + \omega)^2]\hat{r}' = \vec{a}'_{r'} - 2\vec{v}'_{\theta'} \times \vec{\omega} - \omega^2\vec{r}', \\ \vec{a}_\theta &= (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{\theta} = [r'\ddot{\theta}' + 2\dot{r}'(\dot{\theta}' + \omega)]\hat{\theta}' = \vec{a}'_{\theta'} - 2\vec{v}'_{r'} \times \vec{\omega}. \end{aligned} \]
Combining the last two equations, we obtain:
\[ \begin{aligned} m(\vec{a}'_{r'} + \vec{a}'_{\theta'}) &= m(\vec{a}_r + \vec{a}_\theta) + m\omega^2\vec{r}' + 2m(\vec{v}'_{r'} + \vec{v}'_{\theta'}) \times \vec{\omega}, \\ \vec{F}_{\text{eff}} &= \vec{F}_{\text{real}} + \vec{F}_{\text{c}} + \vec{F}_{\text{cor}}. \end{aligned} \]
where \(\vec{F}_{\text{c}}\) and \(\vec{F}_{\text{cor}}\) are the centrifugal force and Coriolis force, respectively:
\[ \begin{aligned} \vec{F}_{\text{c}} &= m\omega^2\vec{r}', \\ \vec{F}_{\text{cor}} &= 2m(\vec{v}'_{r'} + \vec{v}'_{\theta'}) \times \vec{\omega} = 2m\vec{v}' \times \vec{\omega}. \end{aligned} \]