Gravitational elevator (physics Torque question)

[sweetiev]

Homework Statement

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 1.5 m 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. In your trusty Physics book you find that the moment of inertia of a disk is 1/2 that of a ring.

Homework Equations

I think..

Ir=1/2MR^2
Tnet=I(alpha)
Fg-Fr=ma
ar=(Fg-Fr)/Msp

The Attempt at a Solution

I honestly have no idea. Can someone please get me started on it?

Like I have

T1(1.5m)=I(alpha)
T1= [(1/2*m*R^2)(alpha)]/1.5m b/c torque= T1(R)=I*alpha

oh and

there will be T1 and T2,

so Sum of torque = T2(Rp) - T1(Rp) = I*alpha

and... I don't know, I don't understand :(

 

[kuruman]

You need three separate free body diagrams, for the disk, for the pulley and for the hanging cage with people. Use them to write three Newton's 2nd Law equations. Note that you have three unknowns, two tensions and an acceleration. Solve for the acceleration.

 

[kerimek]

I would first commend your curiosity and critical thinking in trying to determine the feasibility of your friend's design. It is always important to consider the physics and mechanics behind any proposed solution.

To get started on this problem, we can first define some variables:
- M = mass of fully loaded elevator
- m = mass of turntable
- R = radius of turntable
- Rp = radius of pulley
- I = moment of inertia of a disk (1/2 MR^2)
- alpha = angular acceleration
- T1 = torque applied by the cable on the turntable
- T2 = torque applied by the cable on the pulley
- Fg = weight of elevator and passengers
- Fr = frictional force between the cable and the turntable
- a = linear acceleration of the elevator

Using the equations you have listed, we can set up the following equations:
1. Torque balance: T2(Rp) - T1(Rp) = I*alpha
2. Net force: Fg - Fr = Ma
3. Angular acceleration and linear acceleration are related by: alpha = a/R

We can also use the given information to determine the values of T1 and T2:
- T1 = (m+M)g, where g is the acceleration due to gravity
- T2 = Mg

Substituting these values into our equations, we get:
1. (Mg)(Rp) - [(m+M)g](Rp) = (1/2 MR^2)(a/R)
2. (Mg) - [(m+M)g] = (M+m)a

Simplifying and rearranging, we get:
1. a = (g/2)(M/m+M)
2. a = (g/2)(M/m+M)

Since we want the acceleration to be much less than g, we can set an upper limit for a, say 0.1g. This means that (M/m+M) should be less than 0.2. This is a feasible value, as the mass of the turntable is twice that of the fully loaded elevator, so (M/m+M) would be around 0.33.

Therefore, based on these calculations, your friend's design seems feasible and could potentially be a viable solution for an emergency escape device. However, it is important to also consider other factors