Chains

Problem:

A rope of length l slides over the edge of a table.  Initially a piece x0 of it hangs without motion over the side of the table.  Let x be the length of rope hanging vertically at time t.  The rope is assumed to be perfectly flexible.
Show that E = T + U is a constant of motion.

Solution:

Problem:

A chain lies pushed together at the edge of a table, except for a piece which hangs over it, initially at rest.  The links of the chain start moving, one at a time.
Show that E = T + U is not a constant of motion.

Solution:

Problem:

A butcher is holding a long string of small-link sausages upright just above the scale pan.  She offers to charge the costumer for just ½ of the maximum reading of the scale after she releases the string.  The customer eagerly agrees.  What percentage of the regular charge does the customer pay?

image

Solution:

Problem:

A uniform, dense rope of length and mass per unit length m is coiled on a smooth table.  You lift one end of the rope by hand vertically upward at constant speed u0.  Find the force that you must apply to the rope when the end is a distance a above the table (a < b). 

Solution:

Problem:

A uniform heavy chain of mass λ per unit length hangs vertically so that the low end just touches a horizontal table.  The upper end is released and the chain falls on the table.  Find the force the chain exerts on the table after it has fallen a distance x.

Solution:

Problem:

A rope of mass M and length L is suspended in the earth's gravitational field, g, with the bottom end of the rope touching a surface.  The rope is released from rest and falls limply on the surface, without bouncing.  Find the force F(t) exerted on the surface as a function of time.  At what time does F(t) reach its maximum?  What is the magnitude of this maximum force?

Solution:

Problem:

A rope with mass M and length L is held in the position shown below, with one end attached to a support. (Assume that only a negligible length of the rope starts out below the support.) The rope is released. Find the force that the support applies to the rope, as a function of time.

image

Solution:

Problem:

A chain with mass/length = m hanging vertically from one end, where an upward force F is applied to it, is lowered onto a table as shown in the figure.
Find the equation of motion for h, the height of the end above the table (h is the length of chain hanging freely).

image

Solution:

Problem:

A ball of mass M on a frictionless horizontal table is pulled by a constant horizontal force F.  A coiled up rope of mass per unit length r is attached which uncoils as the body moves. 
(a)  By taking into account the uncoiling of the rope, derive an expression for the speed of the ball as a function of distance traveled.
(b)  Evaluate the terminal speed of the ball, assuming a very long rope.

Solution: