A chain of length l and mass ë per unit length is fully open and is held on a high frictionless table with y1 of its length hanging from the side of the table as shown in figure. When released, it starts slipping off the table. By calculating the speed of the chain as it slips, show that its energy is conserved
To calculate the speed of the chain as it slips off the table and show that its energy is conserved, we can use the principle of conservation of energy.
Let's consider a small element of length dx from the chain that is still on the table and denoted by x. The gravitational potential energy of this element is given by dU = g(m dx) y2, where g is the acceleration due to gravity, m is the mass of the element dM = ë dx, and y2 is the vertical distance from the center of mass of the element to the reference point at the bottom of the table.
As the chain is released, this small element undergoes horizontal displacement dx and vertical displacement dy = y2 - y1. The work done by the gravitational force on this element is given by dW = -g(m dx) y1 since the force is acting in the opposite direction of displacement.
Using the principle of conservation of energy, we can equate the change in potential energy with the work done:
dU = -dW
g(m dx) y2 = -g(m dx) y1
Dividing both sides by dx and integrating from 0 to l (the length of the chain):
∫₀ˡ g(m dx) y2 = ∫₀ˡ -g(m dx) y1
g(m ∫₀ˡ dx) y2 = -g(m ∫₀ˡ dx) y1
gmy2(∫₀ˡ dx) = -gmy1(∫₀ˡ dx)
gmy2l = -gmy1l
From this equation, we can see that both y2 and y1 have negative signs, indicating that both are negative. This means that y2 = -l and y1 = -y.
Substituting these values back into the equation:
gmy(-l) = -gmyl
Simplifying further:
gml = gml
This equation shows that the chain's energy is conserved since the gravitational potential energy before (left-hand side) and after (right-hand side) it slips off the table are equal.
Now, to calculate the speed of the chain as it slips off the table, we can use the conservation of mechanical energy. The total mechanical energy of the chain is given by the sum of its kinetic energy and potential energy. Since the chain starts from rest, its initial kinetic energy is zero.
At the bottom of the table, all of its initial gravitational potential energy is converted into kinetic energy, which is given by:
mgh = (1/2)mv²
Where m is the mass of the chain, g is the acceleration due to gravity, h is the height of the chain hanging from the table, and v is the speed of the chain at the bottom.
Simplifying the equation, we get:
gh = (1/2)v²
Solving for v, we find:
v = √(2gh)
Therefore, the speed of the chain as it slips off the table is given by v = √(2gh), where h is the height of the chain hanging from the table, and g is the acceleration due to gravity.
This calculation confirms that the chain's energy is conserved since the gravitational potential energy is entirely converted into kinetic energy.