Kinetics is a branch of classical mechanics which studies the relationship between force and motion. Kinematis 은 Kinetics 문제에 속한다. 왜을 묻느냐의 차이인데 Kinematics 는 운동 그 자체 그리고 Kinetics 는 운동의 원인을 묻는다고 보면 된다.
Force laws
이 eq 는 힘의 정의가 아니다. It is not sufficient to describe force as just mass times acceleration. 는 힘의 결과이고 힘 자체는 상호작용의 성질 nature of the interaction 에서 나온다.
Gravitational Force
where,
G &= \text{Gravitational constant}, \\ m_e &= \text{Mass of the Earth}, \\ m_s &= \text{Mass of the Sun}, \\ r &= \text{Distance between the Earth and the Sun}. \end{align}$$ #### Friction Force ##### Drag friction Forces exerted by a fluid medium on a solid object moving through it. This friction force depends on the velocity of the object, its surface properties and medium properties. ##### Dry friction The interaction between two solid objects. ###### no slip $$F_f \leq \mu_s N.$$ Where, $$\begin{align} F_f &= \text{Static friction force}, \\ \mu_s &= \text{Static coefficient of friction}, \\ N &= \text{Normal reaction force}. \end{align}$$ ###### Impending slip $$F_f = \mu_s N.$$ Where, $$\begin{align} F_f &= \text{Static friction force}, \\ \mu_s &= \text{Static coefficient of friction}, \\ N &= \text{Normal reaction force}. \end{align}$$ ###### Slipping $$F_f = \mu_k N.$$ Where, $$\begin{align} F_f &= \text{Kinetic friction force}, \\ \mu_k &= \text{Kinetic coefficient of friction}, \\ N &= \text{Normal reaction force}. \end{align}$$ Most of the times, $\mu_k \leq \mu_s$. #### Linear Elastic Force Linear elastic forces are exerted by objects which have elastic material properties. $$F_e = K \delta$$ Where, $$\begin{align} F_e &= \text{Linear elastic force}, \\ K &= \text{Spring stiffness or spring constant}, \\ \delta &= \text{deformation (stretch or compression)}. \end{align}$$ ### Direction of kinetic friction forces Friction acts in such a way that it opposes the relative motion between two objects. $$v_{A/B} = v_A - v_B$$ $A$ 에 작용하는 마찰은 $v_{A/B}$ 의 반대 운동 마찰은 지면 기준 절대속도가 아니라 상대운동을 방해한다. ### Newton's second law The sum of the forces acting on an object is equal to the mass of that object multiplied by its acceleration. $$\sum \vec{F} = m \vec{a}$$ is valid only when acceleration is measured in an inertial or Newtonian frame, which is a non-accelerating frame of reference. 여기서 $\vec{a}$ 는 반드시 지면 (정지 관측자) 기준 가속도여야 한다. $$\vec{a}_A = \vec{a}_{A/B} + \vec{a}_B$$ 비관성계에서 잰 가속도를 관성계로 바꾸는 공식. ### Newton's second law in different coordinate system #### Cartesian coordinate system $$F_x \ \hat{i} + F_y \ \hat{j} = ma_x \ \hat{i} + ma_y \ \hat{j}$$ - 직선운동, 힘 방향이 고정 - 성분이 서로 독립적 #### Tangent-Normal coordinate system $$F_t \ \hat{e}_t + F_n \ \hat{e}_n = ma_t \ \hat{e}_t + ma_n \ \hat{e}_n = m \dot{v} \ \hat{e}_t + \frac{mv^2}{\rho} \ \hat{e}_n$$ $$\vec{a} = \dot{v} \ \hat{e}_t + \frac{v^2}{\rho} \ \hat{e}_n$$ - 곡선 위를 속력 $v$ 로 움직이고, $\rho$ 를 알때 - $F_t$ 가 속력 변화, $F_n$ 방향 변화 (구심력 요구량) - $\hat{e}_n$ 은 언제나 원 중심 쪽임. - $\hat{e}_b = \hat{e}_t \times \hat{e}_n$ #### Polar coordinate system $$F_r \ \hat{e}_r + F_\theta \ \hat{e}_\theta = ma_r \ \hat{e}_r + ma_\theta \ \hat{e}_\theta = m ( \ddot{r} - r \dot{\theta}^2 ) \hat{e}_r + m (2 \dot{r} \dot{\theta} + r \ddot{\theta}) \hat{e}_\theta$$ $$\vec{a} = ( \ddot{r} - r \dot{\theta}^2 ) \hat{e}_r + (2 \dot{r} \dot{\theta} + r \ddot{\theta}) \hat{e}_\theta$$ - 고정 원점에서 $r, \theta$ 로 기술 - $\hat{e}_r$ 는 반경 바깥, $\hat{e}_\theta$ 는 $\theta$ 증가 방향 - $\ddot{r}$ 반경가속, $- r \dot{\theta}^2$ 구심, $2 \dot{r} \dot{\theta}$ 각가속, $r \ddot{\theta}$ 코리올리 ### General methods for kinetics of particles 1. choose a coordinate system 2. perform the kinematic analysis 3. draw a FBD 4. apply Newton's second law 5. solve for the unknown quantities