Question

You complain about fire safety to the landlord of your high-rise apartment building. he is willing to install an evacuation device if it is cheap and reliable, and he asks you to design it. your proposal is to mount a large wheel (radius 0.400 m) on an axle at its center and wrap a long, light rope around the wheel, with the free end of the rope hanging just past the edge of the roof. residents would evacuate to the roof and, one at a time, grasp the free end of the rope, step off the roof, and be lowered to the ground below. (ignore friction at the axle.) you want a 90.0-kg person to descend with an acceleration of g>4. (a) if the wheel can be treated as a uniform disk, what mass must it have? (b) as the person descends, what is the tension in the rope?

Answer 1

The disk has a moment of inertia of 1/2 M R^2 
and dw/dt = T /I but T = F* R so dw/dt = F*R/I = F*R/( 1/2 M R^2) = 2 F/(MR) Now v = wR so dv/dt = 2 F / M 
For the individual to rush at g/4 then the force performing on the wheel must be 3mg/4 ( where m = 90 kg) so g/4 = 2 * 3mg / 4M 1 = 2*3 m / M M = 6 * 90 = 540kg 
And tension must be 3/4 of the weight of the person = 3mg/4 = 405 N