Displacement of Particle under Force
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Theorem
Let $P$ be a particle of constant mass $m$.
Let the position of $P$ at time $t$ be specified by the position vector $\mathbf r$.
Let a force applied to $P$ be represented by the vector $\mathbf F$.
Then the motion of $P$ can be given by the differential equation:
- $\mathbf F = m \dfrac {\d^2 \mathbf r} {\d t^2}$
or using Newtonian notation:
- $\mathbf F = m \ddot {\mathbf r}$
Proof
\(\ds \mathbf F\) | \(=\) | \(\ds \map {\dfrac \d {\d t} } {m \mathbf v}\) | Newton's Second Law of Motion | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\dfrac \d {\d t} } {m \dfrac {\d \mathbf r} {\d t} }\) | Definition of Velocity | |||||||||||
\(\ds \) | \(=\) | \(\ds m \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf r} {\d t} }\) | Derivative of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds m \dfrac {\d^2 \mathbf r} {\d t^2}\) | Definition of Second Derivative |
$\blacksquare$
Sources
- 1961: D.S. Jones: Electrical & Mechanical Oscillations ... (previous) ... (next): Chapter $1$: Equilibrium: $1.1$ Introduction