Question

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?

Answer 1

part (a). 176580 J

part (b). 197381 J

Step-by-step explanation:

Given,

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

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 =
`\rho dy`

Work done due to the small element,

`dw\ =\ \rho g (60\ -\ y)dy\\`

Total work done to wind the entire chain = w

`w\ =\ \displaystyle\int_(0)^(L) \rho g(60\ -\ y)dy\\\Rightarrow  w\ =\ \rho g\left |(60y\ -\ (y^2)/(2))\ \right |_(0)^(60)\\\Rightarrow w\ =\ 10* 9.81* (60* 60\ -\ (60^2)/(2))\\\Rightarrow w\ =\ 176580\ J`

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

`\therefore W\ =\ w\ +\ mgL\\\Rightarrow W\ =\ 176580\ +\ 35* 9.81* 60\\\Rightarrow W\ =\ 176580\ +\ 20601\\\Rightarrow W\ =\ 197381\ J`