This topic will cover the fundamental concepts of dynamics, including Newton's laws of motion, force, mass, and acceleration. Students will learn how to analyze and solve problems related to the motion of particles and rigid bodies.
Introduction to Dynamics
The study of forces and motion of bodies
| Aspect | Definition |
|---|---|
| Scope | Examines why objects move (cause‑and‑effect) rather than just how they move. |
| Sub‑fields | • Kinematics – description of motion (position, velocity, acceleration). • Kinetics – relation between motion and forces/moments. |
| Contrast | Statics deals with bodies at rest or in uniform motion (net force = 0). Dynamics handles accelerating bodies (net force ≠ 0). |
| Principle | Statement | Typical Equation |
|---|---|---|
| Newton’s First Law (Inertia) | A body remains at rest or moves with constant velocity unless acted upon by a net external force. | ( \sum \mathbf{F}=0 ;\Rightarrow; \mathbf{v}= \text{constant} ) |
| Newton’s Second Law | The rate of change of linear momentum equals the applied net force. | ( \sum \mathbf{F}=m\mathbf{a} ) |
| Newton’s Third Law | For every action there is an equal and opposite reaction. | ( \mathbf{F}{AB}= -\mathbf{F}{BA} ) |
| Work‑Energy Principle | The work done by all forces equals the change in kinetic energy. | ( W_{\text{net}} = \Delta K = \frac12 m v_f^2 - \frac12 m v_i^2 ) |
| Impulse‑Momentum Principle | Impulse applied to a body equals the change in its linear momentum. | ( \mathbf{J}= \int \mathbf{F},dt = \Delta \mathbf{p} ) |
| Angular Momentum | Rate of change of angular momentum equals the applied moment (torque). | ( \sum \mathbf{M}=I\boldsymbol{\alpha} ) |
| Motion | Description | Key Quantities |
|---|---|---|
| Translational | All points of the body move along parallel paths. | Linear displacement (s), velocity (\mathbf{v}), acceleration (\mathbf{a}) |
| Rotational | Motion about a fixed axis; every point follows a circular path. | Angular displacement (\theta), angular velocity (\omega), angular acceleration (\alpha) |
| General Plane Motion | Combination of translation of a reference point and rotation about that point. | Velocity of any point: (\mathbf{v}_P = \mathbf{v}O + \boldsymbol{\omega}\times \mathbf{r}{OP}) |
| Three‑Dimensional Motion | Simultaneous translation and rotation in space. | Use vector forms of (\mathbf{v}, \mathbf{a}, \boldsymbol{\omega}, \boldsymbol{\alpha}) and inertia tensor ( \mathbf{I} ). |
Position → Velocity
[
\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}
]
Velocity → Acceleration
[
\mathbf{a}(t)=\frac{d\mathbf{v}}{dt}= \frac{d^2\mathbf{r}}{dt^2}
]
Constant Acceleration Equations (1‑D