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A 60​-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of 10 ​kg/m. Use 9.8 m divided by s squared for the acceleration due to gravity. a. How much work is required to wind the entire chain onto the cylinder using the​ winch? b. How much work is required to wind the chain onto the cylinder if a 35​-kg block is attached to the end of the​ chain?

Respuesta :

Answer:

part (a). 176580 J

part (b). 197381 J

Explanation:

Given,

  • Density of the chain = [tex]\rho\ =\ 10\ kg/m.[/tex]
  • Length of the chain = L = 60 m
  • Acceleration due to gravity = g = 9.81 [tex]m/s^2[/tex]

part (a)

Let dy be the small element of the chain at a distance of 'y' from the ground.

mass of the small element of the chain = [tex]\rho dy[/tex]

Work done due to the small element,

[tex]dw\ =\ \rho g (60\ -\ y)dy\\[/tex]

Total work done to wind the entire chain = w

[tex]w\ =\ \displaystyle\int_{0}^{L} \rho g(60\ -\ y)dy\\\Rightarrow  w\ =\ \rho g\left |(60y\ -\ \dfrac{y^2}{2})\ \right |_{0}^{60}\\\Rightarrow w\ =\ 10\times 9.81\times (60\times 60\ -\ \dfrac{60^2}{2})\\\Rightarrow w\ =\ 176580\ J[/tex]

part (b)

  • mass of the block connected to the chain = m = 35 kg

Total work done to wind the chain = work done due to the chain + work done due to the mass

[tex]\therefore W\ =\ w\ +\ mgL\\\Rightarrow W\ =\ 176580\ +\ 35\times 9.81\times 60\\\Rightarrow W\ =\ 176580\ +\ 20601\\\Rightarrow W\ =\ 197381\ J[/tex]

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