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

Answer:

part (a). 176580 J

part (b). 197381 J

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

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

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

$\therefore W\ =\ w\ +\ mgL\\\Rightarrow W\ =\ 176580\ +\ 35\times 9.81\times 60\\\Rightarrow W\ =\ 176580\ +\ 20601\\\Rightarrow W\ =\ 197381\ J$