A winch system with a wire rope is designed to handle heavy loads. The following requirements need to be met for efficient and safe operation:

$$ Minimum braking force of the wire rope: not less than $100$ tons $($T$).$

$$ Rope end strength: The rope end fixed to the winch must withstand at least $2$ times the rated pulling force, which is $50$ tons.

$$ Rope end field replaceability: The rope end should be replaceable with minimum shortening of the rope, allowing for quick deployment in case of a failure.

$$ Winch IN conditions:

o Maximum single$-$line pulling force: $50$ tons.

o Rope speed during pulling: $7$ m$/$min$.$

$$ Winch OUT conditions:

o Maximum pull force: $2$ tons.

o Rope speed during pulling: $55$ m$/$min$.$

a$.$ Calculate the required diameter of the wire rope, given that the wire rope material has an ultimate tensile strength of $1800$ MPa. Ensure the rope's minimum braking force is at least $100$ tons.

b$.$ Determine the tensile force on the rope end when the winch is operating at its maximum pulling force of $50$ tons. Verify if the rope end can withstand $2$ times the rated pulling force.

c$.$ Specify the rope attachment method that would allow field replaceability while minimizing the shortening of the rope length, ensuring the system's durability and ease of use.

d$.$ Calculate the power required for the winch to achieve the following

conditions:

o For Winch IN: $50$ tons of force at $7$ m$/$min rope speed.

o For Winch OUT: $2$ tons of force at $55$ m$/$min rope speed.

e$.$ Propose a suitable winch motor and gearbox to meet both the Winch IN

and Winch OUT conditions while ensuring safety, reliability, and

performance under varying loads.