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 r

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)

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Thu Thảo 6 months 2021-08-25T12:22:18+00:00 1 Answers 5 views 0

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    2021-08-25T12:23:41+00:00

    Answer:

    The Height is  H = 70.02 m

    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 = \frac{C}{2 \pi}

                   r = \frac{40,000}{2 \pi } = \frac{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) \ = \frac{change \ in \ radius \ }{2 \pi}

                           = \frac{(40000440-40000*10^3)}{2*3.142}

                           = 70.02m

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