A 900 kg elevator hangs by a steel cable for which the allowable stress is `1.15xx10^(8)" N/m"^(2)`. What is the minimum diameter required if the elevator accelerates upward at `1.5" m/s"^(2)?" Take "g=10" m/s"^(2)`.

Options

A. `(6xx10^(-2))/(sqrt(5pi))m`
B. `(6xx10^(-2))/(sqrt(10pi))m`
C. `(3xx10^(-2))/(sqrt(10pi))m`
D. `(3xx10^(-2))/(sqrt(5 pi))m`

Explanation

To solve the problem of determining the minimum diameter required for a steel cable supporting a 900 kg elevator that accelerates upward at 1.5 m/s², we can follow these steps:

### Step 1: Calculate the total force acting on the elevator
The total force (F) acting on the elevator can be calculated using Newton's second law. The elevator experiences both gravitational force and the force due to its upward acceleration.

\[ F = m(g + a) \]

Where:
- \( m = 900 \, \text{kg} \) (mass of the elevator)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( a = 1.5 \, \text{m/s}^2 \) (upward acceleration)

Substituting the values:

\[ F = 900 \, \text{kg} \times (10 \, \text{m/s}^2 + 1.5 \, \text{m/s}^2) = 900 \times 11.5 = 10350 \, \text{N} \]

### Step 2: Use the allowable stress to find the required cross-sectional area
The allowable stress (\( \sigma \)) is given as \( 1.15 \times 10^8 \, \text{N/m}^2 \). The relationship between stress, force, and area is given by:

\[ \sigma = \frac{F}{A} \]

Rearranging this gives us:

\[ A = \frac{F}{\sigma} \]

Substituting the values we calculated:

\[ A = \frac{10350 \, \text{N}}{1.15 \times 10^8 \, \text{N/m}^2} \approx 8.99 \times 10^{-5} \, \text{m}^2 \]

### Step 3: Relate the cross-sectional area to the diameter
The cross-sectional area \( A \) of a circular cable is given by:

\[ A = \frac{\pi d^2}{4} \]

We can rearrange this to find the diameter \( d \):

\[ d^2 = \frac{4A}{\pi} \]

Substituting the area we found:

\[ d^2 = \frac{4 \times 8.99 \times 10^{-5}}{\pi} \]

Calculating this:

\[ d^2 \approx \frac{3.596 \times 10^{-4}}{3.14159} \approx 1.145 \times 10^{-4} \, \text{m}^2 \]

### Step 4: Calculate the diameter
Now, taking the square root to find \( d \):

\[ d = \sqrt{1.145 \times 10^{-4}} \approx 0.0107 \, \text{m} \approx 10.7 \, \text{mm} \]

### Final Answer
The minimum diameter required for the steel cable is approximately **10.7 mm**.

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