## College Physics (4th Edition)

(a) $a_i = \frac{F_i}{m_i}$ (b) $\tau_i = r_i~F_i$ (c) $a_i = \alpha~r_i$ (d) $\sum_{i=1}^{N}~\tau_i = I~\alpha$
(a) $F_i = m_i~a_i$ $a_i = \frac{F_i}{m_i}$ (b) We can find the torque acting on the particle $m_i$: $\tau_i = r_i~F_i$ (c) $a_i = \alpha~r_i$ (d) We can find the sum of all the torques: $\sum_{i=1}^{N}~\tau_i = \sum_{i=1}^{N}~r_i~F_i$ $\sum_{i=1}^{N}~\tau_i = \sum_{i=1}^{N}~r_i~(m_i~a_i)$ $\sum_{i=1}^{N}~\tau_i = \sum_{i=1}^{N}~r_i~(m_i~\alpha~r_i)$ $\sum_{i=1}^{N}~\tau_i = \sum_{i=1}^{N}~(m_i~r_i^2)~\alpha$ $\sum_{i=1}^{N}~\tau_i = (\sum_{i=1}^{N}~(m_i~r_i^2))~\alpha$ $\sum_{i=1}^{N}~\tau_i = I~\alpha$