The advanced transport system in the distribution center utilizes a network of robotic carts connected by frictionless, massless pulleys and cords as shown. Like a double Atwood machine, this setup involves three robotic carts, E, F, and G, with masses $m_E$ , $m_F$ , and $m_G$ respectively. Determine the acceleration of each robotic cart.

• A. $a_{\(\mathrm{E}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,a_{\(\mathrm{F}\)}=\(\frac{3m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,\(\quad\) a_{\(\mathrm{G}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-3m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g$
• B. $a_{\(\mathrm{E}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}+4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,a_{\(\mathrm{F}\)}=\(\frac{3m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}+4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,\(\quad\) a_{\(\mathrm{G}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-3m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g$
• C. $a_{\(\mathrm{E}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,a_{\(\mathrm{F}\)}=\(\frac{3m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,\(\quad\) a_{\(\mathrm{G}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}+3m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g$
• D. $a_{\(\mathrm{E}\)}=\(\frac{m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,a_{\(\mathrm{F}\)}=\(\frac{3m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}-4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g,\(\quad\) a_{\(\mathrm{G}\)}=\(\frac{-m_{\mathrm{E}\)}m_{\(\mathrm{F}\)}-3m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}}{4m_{\(\mathrm{F}\)}m_{\(\mathrm{G}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{F}\)}+m_{\(\mathrm{E}\)}m_{\(\mathrm{G}\)}}g$