High School Climbing Rope: Physics and Forces on Ceiling

[Shardul Khare]

When a perosn is climbing up a rope with mass a ( attached to ceiling) with acceleration a how does the physics work and what forces will be applied on ceiling?
I know when the rope is massless all the force gets transferred ,but now hwat will happen

 

[BvU]

Let me first propose to use $m_{\rm rope}$ instead of a for the mass of the rope (it isn't wise to use the same symbol for two different things, to put it mildly).

The ceiling has to exert the same force as when the rope is massles, plus the constant force $m_{\rm rope} \; g$. (the rope does not accelerate or anything)