Question

Consider a 40,000 km steel pipe in the shape of a ring that fits snuggly all around the circumference of the Earth. We are heating now the ring so its temperature increased by 1 degree C. Now, the pipe will no longer be snug. How high the ring will now stand above ground level? (Make as many simplifications as necessary). Data: Coefficient of linear expansion for steel is 11*10-6 /degree C. This means, for example, that a 1-meter bar of steel that increases its temperature by 1 degree C will expand 11*10-6 meters (11 micrometers)

Answer 1

The Height is H = 70.02 m

Step-by-step explanation:

We are given that the

Initial length is =
`40000\ Km` =
`40,000 *10^(3) m`

from what we are told in the question the circumference of the circle is =
`40,000 Km`

This means that the Radius would be :

Let C denote the circumference

So

`C = 2 \pi r`

=>
`r = (C)/(2 \pi)`

`r = (40,000)/(2 \pi ) = (40,000*10^(3))/(2 *3.142) = 6.365*10^6 m`

We are told that 1-meter bar of steel that increases its temperature by 1 degree C will expand
`11*10^(-6)` meters

Hence

The final length would be

`40000*10^3 *(T + \alpha )`

Where T is the change in temperature
`\alpha` is the Coefficient of linear expansion for steel

let
`L_(final)` denote the final length

So

`L_(final) =40000*10^(6) *[1+ 11*10^(-6)]`

`= 40000440 \ m`

Now the Height is mathematically represented as

`Height(H) \ = (change \ in \ radius \ )/(2 \pi)`

`= ((40000440-40000*10^3))/(2*3.142)`

`= 70.02m`