Kinematics and Particle Dynamics

Kinematics describes the geometry of motion, which is fundamentally relative to an observer's choice of reference frame.

1. Inertial Frames and the Principle of Galilean Relativity

An **inertial frame** is a reference frame in which Newton's first law holds: an isolated body free from external forces moves with constant velocity in a straight line. If $S$ is an inertial frame and $S'$ moves relative to $S$ with constant translational velocity $\mathbf{V}_0$: $$\mathbf{r}'(t) = \mathbf{r}(t) - \mathbf{V}_0 t, \quad t' = t$$ Differentiating with respect to time: $$\mathbf{v}' = \mathbf{v} - \mathbf{V}_0, \quad \mathbf{a}' = \frac{d\mathbf{v}'}{dt} = \frac{d\mathbf{v}}{dt} = \mathbf{a}$$ Because acceleration is identical in all inertial frames, Newton's second law $\mathbf{F} = m\mathbf{a}$ retains the exact same mathematical form in all inertial frames. This is the **Principle of Galilean Relativity**.

2. Linearly Accelerating Reference Frames

If frame $S'$ has translational acceleration $\mathbf{A}_0(t)$ relative to inertial frame $S$: $$\mathbf{a}' = \mathbf{a} - \mathbf{A}_0$$ Multiplying by particle mass $m$: $$m \mathbf{a}' = m \mathbf{a} - m \mathbf{A}_0 = \mathbf{F}_{\text{real}} + \mathbf{F}_{\text{fictitious}}$$ where $\mathbf{F}_{\text{fictitious}} = -m\mathbf{A}_0$ is an inertial (pseudo) force that must be added to preserve Newton's second law in the accelerating frame.

3. Rotating Reference Frames: Centrifugal and Coriolis Forces

If frame $S'$ rotates with angular velocity $\boldsymbol{\omega}$ about an axis passing through the origin of inertial frame $S$: The time derivative of any vector $\mathbf{Q}$ relates by the operator identity: $$\left(\frac{d\mathbf{Q}}{dt}\right)_S = \left(\frac{d\mathbf{Q}}{dt}\right)_{S'} + \boldsymbol{\omega} \times \mathbf{Q}$$ Applying this twice to the position vector $\mathbf{r}$ yields the exact transformation of accelerations: $$\mathbf{a}_S = \mathbf{a}_{S'} + 2(\boldsymbol{\omega} \times \mathbf{v}_{S'}) + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}) + \dot{\boldsymbol{\omega}} \times \mathbf{r}$$ The effective equation of motion in the rotating frame is: $$m \mathbf{a}_{S'} = \mathbf{F}_{\text{real}} + \mathbf{F}_{\text{coriolis}} + \mathbf{F}_{\text{centrifugal}} + \mathbf{F}_{\text{euler}}$$ where:

• Coriolis Force: $\mathbf{F}_{\text{coriolis}} = -2m(\boldsymbol{\omega} \times \mathbf{v}_{S'})$. Acts perpendicular to velocity; deflects winds rightward in the Northern Hemisphere (cyclones).
• Centrifugal Force: $\mathbf{F}_{\text{centrifugal}} = -m\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}) = m\omega^2 \mathbf{r}_\perp$. Acts radially outward from the rotation axis.
• Euler Force: $\mathbf{F}_{\text{euler}} = -m(\dot{\boldsymbol{\omega}} \times \mathbf{r})$. Arises only when angular acceleration is non-zero.

For general curvilinear motion along an arbitrary planar trajectory, intrinsic coordinates based on arc length $s(t)$ separate speed changes from directional changes.

1. Intrinsic Unit Basis Vectors

Let $\hat{\mathbf{t}}$ be the unit tangent vector pointing along velocity $\mathbf{v}$, and let $\hat{\mathbf{n}}$ be the principal unit normal vector pointing toward the local center of curvature: $$\mathbf{v} = v \hat{\mathbf{t}}, \quad v = \frac{ds}{dt} = \dot{s}$$ Differentiating with respect to time: $$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{dv}{dt} \hat{\mathbf{t}} + v \frac{d\hat{\mathbf{t}}}{dt}$$ Using the geometric Frenet-Serret relation $\frac{d\hat{\mathbf{t}}}{ds} = \frac{1}{\rho} \hat{\mathbf{n}}$, where $\rho$ is the local radius of curvature: $$\frac{d\hat{\mathbf{t}}}{dt} = \frac{d\hat{\mathbf{t}}}{ds}\frac{ds}{dt} = \left(\frac{1}{\rho}\hat{\mathbf{n}}\right) v = \frac{v}{\rho} \hat{\mathbf{n}}$$ Substituting into the total acceleration: $$\mathbf{a} = a_t \hat{\mathbf{t}} + a_n \hat{\mathbf{n}} = \left( \frac{dv}{dt} \right) \hat{\mathbf{t}} + \left( \frac{v^2}{\rho} \right) \hat{\mathbf{n}}$$

2. Dynamical Interpretation

• Tangential Component ($a_t = \dot{v} = \ddot{s}$): Governs changes in the *magnitude* of velocity (speed). Caused by net forces parallel to trajectory: $F_t = m a_t$.
• Normal (Centripetal) Component ($a_n = \frac{v^2}{\rho}$): Governs changes in the *direction* of velocity. Caused by net lateral forces perpendicular to trajectory: $F_n = m \frac{v^2}{\rho}$.
• Total Acceleration Magnitude: $$a = \sqrt{a_t^2 + a_n^2} = \sqrt{\left(\frac{dv}{dt}\right)^2 + \left(\frac{v^2}{\rho}\right)^2}$$
• Radius of Curvature Formula: For a path expressed as $y = f(x)$: $$\rho(x) = \frac{[1 + (y')^2]^{3/2}}{|y''|}$$

Two-dimensional ballistic motion under constant gravitational acceleration provides the classical prototype of particle kinematics.

1. Kinematic Equations on Flat Terrain

For a particle launched from $(0, 0)$ with initial speed $v_0$ at angle $\theta_0$ to horizontal: $$a_x = 0, \quad a_y = -g$$ Integrating: $$v_x(t) = v_0 \cos \theta_0, \quad v_y(t) = v_0 \sin \theta_0 - gt$$ $$x(t) = (v_0 \cos \theta_0) t, \quad y(t) = (v_0 \sin \theta_0) t - \frac{1}{2}gt^2$$ Eliminating $t$ yields the parabolic trajectory: $$y(x) = x \tan \theta_0 - \frac{g}{2 v_0^2 \cos^2 \theta_0} x^2$$ Trajectory metrics:

• Time of Flight ($T$): $T = \frac{2 v_0 \sin \theta_0}{g}$.
• Maximum Height ($H$): $H = \frac{v_0^2 \sin^2 \theta_0}{2g}$.
• Horizontal Range ($R$): $R = \frac{v_0^2 \sin 2\theta_0}{g}$, maximized at $\theta_0 = 45^\circ$.

2. Projectile on an Inclined Plane

When a projectile is launched at angle $\alpha$ relative to horizontal onto a hill inclined at angle $\beta$ ($alpha > \beta$): Rotating axes so that $x'$ is along the slope and $y'$ is normal to the slope: $$a_{x'} = -g \sin \beta, \quad a_{y'} = -g \cos \beta$$ $$v_{0x'} = v_0 \cos(\alpha - \beta), \quad v_{0y'} = v_0 \sin(\alpha - \beta)$$ Solving $y'(T) = 0$ gives time of flight: $$T = \frac{2 v_0 \sin(\alpha - \beta)}{g \cos \beta}$$ Range along the inclined slope: $$R_{\text{incline}} = x'(T) = \frac{2 v_0^2 \cos \alpha \sin(\alpha - \beta)}{g \cos^2 \beta}$$ Maximized when the launch angle bisects the remaining angle: $\alpha_{\text{opt}} = \frac{\pi}{4} + \frac{\beta}{2}$.

Circular motion occurs when a particle moves along a circular path of fixed radius $R$.

1. Uniform Circular Motion

If speed $v$ is constant: $$a_t = \frac{dv}{dt} = 0, \quad a_n = \frac{v^2}{R} = \omega^2 R$$ The acceleration is purely radial (centripetal), pointing toward the center. By Newton's second law: $$F_c = m \frac{v^2}{R} = m \omega^2 R$$

2. Motion in a Vertical Circle

When a particle of mass $m$ is attached to a string of length $R$ and swung in a vertical plane under gravity: By energy conservation between bottom (speed $v_0$) and angle $\theta$ from bottom: $$\frac{1}{2}m v_0^2 = \frac{1}{2}m v^2 + mgR(1 - \cos \theta) \implies v^2 = v_0^2 - 2gR(1 - \cos \theta)$$ Newton's second law along the radial direction gives string tension $T$: $$T - mg \cos \theta = \frac{mv^2}{R} \implies T = mg \cos \theta + \frac{m}{R}[v_0^2 - 2gR(1 - \cos \theta)]$$ $$T(\theta) = \frac{m v_0^2}{R} - mg(2 - 3\cos \theta)$$ Critical Conditions:

• At top of loop ($\theta = \pi$): $T_{\text{top}} = \frac{m v_0^2}{R} - 5mg$. For the string not to go slack ($T_{\text{top}} \ge 0$): $$v_{\text{top}} \ge \sqrt{gR}, \quad v_0 \ge \sqrt{5gR}$$
• Difference in tension between bottom and top is always independent of launch speed: $$T_{\text{bottom}} - T_{\text{top}} = 6mg$$

Newtonian particle dynamics relates external forces to resulting particle trajectories.

1. Newton’s Three Laws

1. First Law (Inertia): A body remains in rest or uniform straight-line motion unless acted upon by a net force: $\sum \mathbf{F} = \mathbf{0} \implies \mathbf{v} = \text{const}$.
2. Second Law (Dynamical Evolution): Net force equals time rate of change of linear momentum $\mathbf{p} = m\mathbf{v}$: $$\mathbf{F} = \frac{d\mathbf{p}}{dt} = m \frac{d\mathbf{v}}{dt} + \mathbf{v} \frac{dm}{dt}$$ For constant mass: $\mathbf{F} = m \mathbf{a}$.
3. Third Law (Reciprocity): Pairwise mutual interaction forces between bodies $A$ and $B$ are collinear, equal in magnitude, and opposite in direction: $\mathbf{F}_{AB} = -\mathbf{F}_{BA}$.

2. Laws of Dry Friction (Coulomb-Amontons)

Contact forces parallel to surfaces arise from microscopic roughness and molecular bonding:

• Static Friction ($f_s$): Self-adjusting force opposing applied force up to a maximum threshold: $$f_s \le \mu_s N$$
• Kinetic Friction ($f_k$): Dynamic resistive force during relative sliding: $$f_k = \mu_k N, \quad \text{with } \mu_k < \mu_s$$
• Angle of Friction ($\lambda$) and Angle of Repose ($\theta_r$): The maximum static friction angle satisfies $\tan \lambda = \mu_s$. On an inclined plane, a block begins sliding under gravity when slope exceeds the angle of repose $\theta_r = \arctan(\mu_s)$.