How Can Friction Impact the Movement of a Chain Off a Table?

[gaborfk]

A uniform chain of length 8.00m initially lies stretched out on a horizontal table.

a) If the coefficient of static friction between the chain and the table is 0.600, show that the chain will begin to slide off the table if at least 3.00m of it hangs over the edge of the table.

b)
Determine the speed of the chain as all of it leaves the table, given that the coefficient of the kinetic friction between the chain and the table is 0.400.


I have solved part a no problem. Part b I am having a lot of problems with. I am trying to fill in the equation U(init)+K(init)-W(fric)=U(final)+K(final)


I understand that the right side is U(final) is 0 and K(final) is 1/2mv^2. How about the left side?

Thank you in advance

 

[Doc Al]

Don't you have a thread started on this same problem already?
https://www.physicsforums.com/showthread.php?t=48928

Here are some hints. If U(final) is zero, where are you measuring PE from? Use that same point to measure the initial PE from. Doesn't the chain start falling from rest? Figure out the work done against friction.

 

[wais]

for any help!

For part b, the left side of the equation represents the initial potential and kinetic energy of the chain, while the right side represents the final kinetic energy of the chain as it leaves the table.

To solve for the initial potential and kinetic energy, we need to first calculate the height of the chain that is still on the table. We know that the total length of the chain is 8.00m and at least 3.00m of it hangs over the edge of the table, leaving 5.00m of the chain still on the table.

Using the formula for potential energy, U=mgh, we can calculate the potential energy of the 5.00m of chain on the table. The mass of the chain can be calculated using its linear density (mass per unit length) and the length of the chain on the table.

Next, we need to calculate the initial kinetic energy of the chain. Since the chain is initially at rest on the table, its initial kinetic energy is 0.

Now, we can plug these values into the left side of the equation:

U(init)+K(init)-W(fric) = mgh + 0 - W(fric)

The work done by friction, W(fric), can be calculated using the formula W = Fd, where F is the force of friction and d is the distance over which the force is applied. In this case, the force of friction is equal to the coefficient of kinetic friction (0.400) multiplied by the weight of the chain (mg) and the distance over which it acts is the length of the chain (8.00m).

Once you have calculated the work done by friction, you can plug it back into the equation and solve for the initial kinetic energy. Once you have the initial kinetic energy, you can solve for the speed using the formula K(final) = 1/2mv^2.

Hope this helps!