Show your work and reasoning for below.
After watching a news story about a fire in a high rise apartment building, you and your friend decide to design an emergency escape device from the top of a building.
To avoid engine failure, your friend suggests a gravitational powered elevator. The design has a large, heavy turntable (a horizontal disk that is free to rotate about its center) on the roof with a cable wound around its edge. The free end of the cable goes horizontally to the edge of the building roof, passes over a heavy vertical pulley, and then hangs straight down.
A strong wire cage which can hold 5 people is then attached to the hanging end of the cable. When people enter the cage and release it, the cable unrolls from the turntable lowering the people safely to the ground.
To see if this design is feasible you decide to calculate the acceleration of the fully loaded elevator to make sure it is much less than "g."
Your friend's design has the radius of the turntable disk as 1.5m and its mass is twice that of the fully loaded elevator. The disk which serves as the vertical pulley has 1/4 the radius of the turntable and 1/16 its mass. The moment of inertia of the disk is 1/2 that of a ring.

Explanation:

Given that,

His friend design has a turnable disk of radius 1.5m

R = 1.5m

The mass is twice the fully loaded elevator.

Let the mass of the full loaded elevator be M

Then, mass of the turn able

Mt = ½M

Radius of the disk that serves as a vertical pulley is ¼ radius of turntable and 1/16 of the mass.

Mass of pulley is

Mp = 1 / 16 × Mt

Mp = 1 / 16 × M / 2

Mp = M / 32

Also, radius of pulley

Rp = ¼ × R = ¼ × 1.5

Rp = 0.375m

The moment of inertia of the disk of a ring is

I = ½MR²

To calculate the moment of the turntable, we can use the formula

I_t = ½Mt•R²

I_t = ½ × ½M × 1.5²

I_t = 0.5625 M

I_t = 9M / 16

Also, the moment of inertia of the vertical pulley

I_p = ½Mp•Rp

I_p = ½ × (M/16) × 0.375

I_p = 0.01171875 M

I_p = 3M / 256

Let assume that, the tension in the cable between the pulley and the elevator is T1 and Let T2 be the tension between the turntable and the pulley

So, applying newton second law of motion,

For the elevator

Fnet = ma

Mg - T1 = Ma

a = (Mg-T1) / M

For vertical pulley,

The torque is given as

τ_p = I_p × α_p = (T2—T1)•r

τ_p = 3M/256 × α_p = (T2-T1)•r

For turntable

The torque is given as

τ_t = I_t × α_t = T2•r

τ_t = 9M/16 × α_t = T2•r

So, the torque are equal

τ_t = τ_p

9M/16 × α_t = 3M/256 × α_p

M cancel out

9•α_t / 16 = 3•α_p / 256

Cross multiply

9•α_t × 256 = 3•α_p × 16

Divide both sides by 48

48•α_t = α_p

α_t = α_p / 48

Then, from,

τ_t = 9M/16 × α_t = T2•r

T2•r = 9M / 16 × α_p / 48

T2•r = 3Mα_p / 16

Also, from

τ_p = 3M/256 × α_p = (T2-T1)•r

3M/256 × α_p = T2•r - T1•r

T1•r = T2•r - 3M/256 × α_p

T1•r = 3Mα_p / 16 - 3M/256 × α_p

T1•r = 3Mα_p / 16 - 3Mα_p/256

T1•r = 45Mα_p / 256

T1 = 45Mα_p / 256R

Then, from

a = (Mg-T1) / M

a = Mg - (45Mα_p / 256R) / M

a = g - 45α_p / 256

From the final answer, it is show that the acceleration is always less than acceleration due to gravity due to the subtraction of 45α_p / 256 from g