- #1

Satvik Pandey

- 591

- 12

## Homework Statement

There is a chain of uniform density on a table with negligible friction. The length of the entire chain is 1 m. Initially, one-third of the chain is hanging over the edge of the table. How long will it take the chain (in seconds) to slide off the table?

## Homework Equations

## The Attempt at a Solution

I tried to use conservation of energy.

Initially position of CM of hanging part of chain wrt to table is 1/3.

And mass of part of chain hanging down the table is 2m/3.

So ##{ E }_{ pi }=\frac { 2mg }{ 9 } ##

If the chain falls by distance '##x##'

then position of CM of hanging part of chain wrt to table will be ##(3x+2)/6##

And mass of part of chain hanging down the table will be ##(3x+2)m/3##

So ##{ E }_{ px }=\frac { { (2+3x) }^{ 2 } }{ 18 } mg##

The potential energy is converted into kinetic energy.

So ##\frac { { (2+3x) }^{ 2 } }{ 18 } mg-\frac { 2mg }{ 9 } =\frac { 1 }{ 2 } \times \frac { (2+3x)m }{ 3 } { v }^{ 2 }##

on simplifying I got

##\frac { { (2+3x) }^{ 2 } }{ 18 } mg-\frac { 2mg }{ 9 } =\frac { 1 }{ 2 } \times \frac { (2+3x)m }{ 3 } { v }^{ 2 }##

or ##\frac { 1 }{ { v }^{ 2 } } =\frac { (3x+2) }{ x(3x+4)g } ##

or ##\int { dt } =\int { \sqrt { \frac { (3x+2) }{ x(3x+4)g } } dx } ##

Is this correct till here?