Question

A chain of mass m and length l lies on a horizontal table. The chain is allowed to slide down gently from the side of the table. Find the speed of the chain at the instant when the last link of the chain slides from the table. Neglect friction everywhere.
A) $\sqrt {gl}$
B) $\sqrt {2gl}$
C) $\sqrt {3gl}$
D) $\sqrt {5gl}$

Answer 1

The potential energy of the chain can be calculated when it falls down the table and then equated to kinetic energy because according to the law of conservation of energy, both will be equal. From the equation, the value of speed can be calculated in terms of g and l only.

Formula used:
Potential energy (PE) = mgh where,
m = mass of the object
g = acceleration due to gravity
h = height of the object
Kinetic energy (KE) = $\dfrac{1}{2}m{v^2}$ where,
m = mass of the object
v = velocity of the object

Complete step by step answer:
Given:
Mass of chain = m
Length of chain = l
1. When the chain is on the table

2. When the chain slides down

If we consider the potential energy of the chain in 1. with respect to the table top, it is 0 because:
It is placed on the table top and its height will thus be neglected.
$P{E_1}$= mgh
$\Rightarrow P{E_1}$ = 0 ; if h = 0.
In the second case, the chain is falling from the table, the height from the table top will be equal to the centre of mass of the rod (so as to have an average height for the respective mass)
If length of rod is l, its centre of mass will be half its length:
$h = \dfrac{l}{2}$
$\Rightarrow P{E_2} = mg\dfrac{l}{2}$
Kinetic energy Is given as:
$KE = \dfrac{1}{2}m{v^2}$
Now, according to the law of conservation energy, when an object falls from a height it’s decrease in potential energy is equal to the increase in kinetic energy.
Decrease in potential energy (P) is:
P = $P{E_2}$ - $P{E_1}$

$$PE = mg\dfrac{l}{2} - 0 \\\ \Rightarrow PE = mg\dfrac{l}{2} \\\$$

$\Rightarrow mg\dfrac{l}{2} = \dfrac{1}{2}m{v^2}$
Calculating the value of v:
$mg\dfrac{l}{2} = \dfrac{1}{2}m{v^2} \\\ \Rightarrow{v^2} = gl \\\ \therefore v = \sqrt {gl} \\\$
The speed of the chain at the instant when the last link of the chain slides from the table neglecting friction everywhere is $\sqrt {gl}$ and the correct option is A).

Note: Everybody has a different mass distribution, the centre of mass is that point where the whole mass of the body is supposed to be concentrated and thus when we talk about a falling body, we generally consider its centre of mass only. The law of conservation of energy states that the energies transform from one form to another (like here from potential to kinetic) but can neither be created nor be destroyed. An object possesses potential energy due to its position and kinetic energy due to its motion.