Physics - Dynamics

Dynamics

Introduction

Newton's Second Law

$$\begin{equation} \mathbf{F} = m\mathbf{a}= m \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2} \end{equation}$$

Newton's Third Law

The force exerted by object $i$ onto object $j$ is opposite of the force exerted by object $j$ onto object $i$. (We use lower case $f$ for internal forces):

$$\begin{equation} \mathbf{f}_{ij} = - \mathbf{f}_{ji} \end{equation}$$

External Force

Denote the total external force exerted on object $i$ as (we use upper case $F$ for external forces) $\mathbf{F}_i$.

Applications

One object without external force

$\mathbf{F} = 0$, therefore, $\frac{d\mathbf{v}}{dt} = 0$, i.e., $\mathbf{v}$ does not change. The trajectory of the object is:

$\mathbf{x} = \mathbf{x}_{0} + \mathbf{v}t$.

One object with constant force (same direction)

$\mathbf{a} = \frac{\mathbf{F}}{m}$

$\mathbf{x} = \mathbf{x}_0 + \mathbf{v}_0t + \frac{1}{2}\mathbf{a}t^2$

Average velocity:

$\bar{\mathbf{v}} = \frac{1}{2}(\mathbf{v}_0 + \mathbf{v}_f)$

Time spent:

$t = \frac{\mathbf{v}_f - \mathbf{v}_0}{\mathbf{a}}$

Total distance:

$\mathbf{S} = \bar{\mathbf{v}}t= \frac{1}{2}(\mathbf{v}^2_f - \mathbf{v}^2_0)$

Two objects without external force

$$\mathbf{f}_{ji} = m_i \frac{d\mathbf{v}_i}{dt}$$

$$\mathbf{f}_{ij} = m_j \frac{d\mathbf{v}_j}{dt}$$

The sum gives (uses Equation (2)):

$$\begin{equation} m_i \frac{d\mathbf{v}_i}{dt} + m_j \frac{d\mathbf{v}_j}{dt} = 0 \end{equation}$$

Rewritten as:

$$\frac{d (m_i \mathbf{v}_i + m_j \mathbf{v}_j)}{dt} = 0$$

Conservation of momentum:

$$\begin{equation} m_i \mathbf{v}_i + m_j \mathbf{v}_j = const \end{equation}$$

Equation (3) can also be written as:

$$\begin{equation} m_i \frac{d^2\mathbf{x}_i}{dt^2} + m_j \frac{d^2\mathbf{x}_j}{dt^2} =
\frac{d^2 (m_i \mathbf{x}_i + m_j \mathbf{x}_j)}{dt^2} = 0 \end{equation}$$

If we define the center of mass as:

$$\begin{equation} \mathbf{x}_{c} = \frac{m_{i}}{m_{i}+m_{j}}\mathbf{x}_i + \frac{m_{j}}{m_{i}+m_{j}}\mathbf{x}_j \end{equation}$$

Equation (5) becomes:

$$\begin{equation} (m_i+m_j) \frac{d^2 (\frac{m_i}{m_i+m_j}\mathbf{x}_i + \frac{m_j}{m_i+m_j} \mathbf{x}_j)}{dt^2} = (m_i+m_j) \frac{d^2 (m_i+m_j)\mathbf{x}_c}{dt^2} = 0 \end{equation}$$

Therefore, we can view the two objects as one object with mass $m_i+m_j$ and their center of mass $\mathbf{x}_c$ moves with a constant velocity.

If there are external forces, we can repeat the process, the result is as if the total external forces act on the center of mass of the whole system.