An interesting fire escape possibility involves a platform attached to a rope th

Question

An interesting fire escape possibility involves a platform attached to a rope that is wound around a massive, solid cylindrical flywheel that is free to rotate about its central axis. (See the not-to-scale diagram below.) A person steps out of a window onto the platform and falls at increasing speed to the ground below, causing the flywheel to angularly accelerate at the same time. The massive flywheel constrains the rate at which the person falls to the ground. Assume a safe landing speed to be 2.00 m/s on the ground, 10.0 m below the starting position. Your task is to determine the flywheel mass needed, using three different techniques. The person and platform mass is 80.0 kg; the flywheel's radius is 0.800 m. Solve for mass of flywheel using the law of conservation of energy. Solve for mass of flywheel by combining Newton's 2^nd law for translational motion, F_NET = ma, for the falling mass and Newton's 2^nd law for rotational motion, tau_NET = I_a, for the rotating flywheel. Solve for mass of flywheel using Newton's 2^nd law for rotational motion, tau_NET = dL/dt for both the flywheel and the falling mass.

Explanation / Answer

a) Work done by gravity = change in KE of system

80 x 9.81 x 10 = [ (80 x 2^2 /2 ) + ((m x 0.8^2 /2) (2/0.8)^2 /2 ) ]

kientic energy of flywheel = Iw^2 /2

I = mR^2 /2 and w = v/R

m = 7688 kg

b) using v^2 - u^2 = 2as

2^2 - 0 = 2 x a x 10

a = 0.2 m/s^2

Using F = ma on platform,

80g - T = 80a

T = 80 x (9.8 - 0.2) = 768 N

on wheel, torque = I x alpha

alpha = a/R

R x T = (mR^2 /2 ) (a/R)

T = ma/2

768 = m ( 0.2)/2

m = 7680 kg

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