Dinamica del corpo rigido e attrito

$\sum_{ }^{ }\overrightarrow{F_e}+m\overrightarrow{a_G}\ =\ 0$

$\sum_{ }^{ }\overrightarrow{F_e}+\overrightarrow{F_{inerzia}}\ =\ m\overrightarrow{a_G}$

$\overrightarrow{F_{inerzia\ }}\ =\ -m\overrightarrow{a_G}$

$\sum_{ }^{ }\overrightarrow{F_e}+\overrightarrow{F_{inerzia}}\ =\ 0$

$$\sum_{ }^{ }\overrightarrow{M_e^{\left(P\right)}}\ =\left(G-P\right)\times m\overrightarrow{a_P}$$  

$$\sum_{ }^{ }\overrightarrow{M_e^{\left(P\right)}}\ =\ J_G\frac{\text{d}\overrightarrow{\omega}}{\text{d}t}$$  

$$\sum_{ }^{ }\overrightarrow{M_e^{\left(P\right)}}\ =\ J_G\frac{\text{d}\overrightarrow{\omega}}{\text{d}t}+\left(G-P\right)\times m\overrightarrow{a_P}$$  

$$\sum_{ }^{ }\overrightarrow{M_e^{\left(P\right)}}\ -\ J_G\frac{\text{d}\overrightarrow{\omega}}{\text{d}t}-\left(G-P\right)\times m\overrightarrow{a_G}\ =\ 0$$  

$$\left|\overrightarrow{F}\right|\le f_s\ mg$$

$$\left|\overrightarrow{F}\right|\ge f_s\ mg$$

$$\left|\overrightarrow{F}\right|\le f_d\ mg$$

$$\left|\overrightarrow{F}\right|\le f_s\left(mg-ma\right)$$

$$\overrightarrow{T_2\ }\ =\ f_d\ m_{2\ }g\ \frac{\overrightarrow{v_2}-\overrightarrow{v_1}}{\left|\overrightarrow{v_2}-\overrightarrow{v_1}\right|}$$  

$$\overrightarrow{T_2\ }\ =\ f_d\ m_{2\ }g\ \frac{\overrightarrow{v_1}-\overrightarrow{v_2}}{\left|\overrightarrow{v_1}-\overrightarrow{v_2}\right|}$$  

$$\left|\overrightarrow{T_1}\right|\ \ =\ f_d\ m_{2\ }g\ \frac{\left|\overrightarrow{v_1}\right|+\left|\overrightarrow{v_2}\right|}{\left|\overrightarrow{v_1}-\overrightarrow{v_2}\right|}$$  

$$\overrightarrow{T_1\ }\ =\ -f_d\ m_{2\ }g\ \frac{\overrightarrow{v_2}}{\left|\overrightarrow{v_2}\right|}$$  

$$\left|\overrightarrow{F}\right|\ =\ m\frac{\text{d}\omega}{\text{d}t}R$$  

$$\left|\overrightarrow{F}\right|\ =\ m\frac{\text{d}\omega}{\text{d}t}R\ +\frac{J\frac{\text{d}\omega}{\text{d}t}}{R}$$  

$$\left|F\ -m\frac{\text{d}\omega}{\text{d}t}R\right|\ \le\ f_s\ m\ g$$  

$$\left|\overrightarrow{F}\right|\ =\ -m\frac{\text{d}\omega}{\text{d}t}R$$  

$$\Pi^{\left(pressione\right)}\ =\ \left(p\ S\ \overrightarrow{\ i\ '}\right)\ \circ\ \overrightarrow{v_{B,\ rel}}$$  

$$\Pi^{\left(attrito\right)}\ =\ -m\ g\ f_d\left|\overrightarrow{v_C}\right|$$  

$$\Pi^{\left(peso\right)}\ =\ 0$$  

$$\frac{\text{d}E_c}{\text{d}t}\ =\ m\ \left|\overrightarrow{v_C}\right|\left|\overrightarrow{a_C}\right|$$  

$\Pi^{\left(e\right)}\ =\ M\ \omega$

$\left|\frac{M}{r}\right|\le f_s\left|m\ g\right|$

$\frac{\text{d}E_c}{\text{d}t}\ =\ m\left|\overrightarrow{v_C}\right|\left|\overrightarrow{a_C}\right|$

$\Pi^{\left(attrito\right)}\ =\ -f_s\left|\overrightarrow{N}\right|\left|\overrightarrow{v_C}\right|$

è dovuto all'isteresi del materiale

dissipa potenza a causa dei micro-slittamenti

non può esserci se c'è attrito statico

è causato da microsaldature tra i due corpi

$$\left|W_{diss}\right|\ =\ f_s\ m\ g\ \left|\overrightarrow{v_C}\right|+f_vm\ g\ \omega\ R$$

$$\left|\overrightarrow{T}\right|\ =\ f_v\ m\ g$$

$$\left|W_{diss}\right|\ =\ f_vm\ g\ \omega\ R$$

$$\left|\overrightarrow{T}\right|\ =\ f_s\ m\ g$$

sono causati dall'isteresi del materiale

la zona di aderenza si estende al crescere della forza T

$$\left|W_{diss}\right|\ =\ \left|\overrightarrow{T}\right|\ \left|\overrightarrow{v_C}\right|\ \epsilon$$

$$\left|W_{diss}\right|\ =\ \left|\overrightarrow{T}\right|\ \omega\ R\ \epsilon$$