Consider the system shown, in which two masses are on a frictionless ramp and one mass is hanging from a massless, inextensible string. Let what is labeled $m_1=3m,m_2=m,$ and $m_3=m,$ where $$ is an unknown coefficient. Considering $m$ and $$ to be known, find expressions for:

a$)$ The tension in the rope if the system is in equilibrium.

b$)$ The acceleration of the system if the system is in equilibrium. Hint: just think about this one.

c$)$ The coefficient $.$

d$)$ The force of $m_1$ on $m_2$ if the system is in equilibrium.

Now, let $$ suddenly become half of what you found in c$).$ This will put the system out of equilibrium. Obtain expressions for:

e$)$ The new tension in the rope.

f$)$ The new acceleration of the system.

g$)$ The new force of $m_1$ on $m_2.$