A uniform chain of total mass m is laid out straight on a frictionless table and held stationary so that one-quarter of its length, L = 1.79 m, is hanging vertically over the edge of the table. The chain is then released. Determine the speed of the chain at the instant when only one-quarter of its length remains on the table.
The easy way to do this is with conservation of energy, since you have to deal with an increasing acceleration rate, which would make the equation of motion rather messy. You can use energy conservation because it is frictionless.
When it starts out, the center of mass of the chain is (1/2)*(L/4) = L/8 below the table surface.
When 1/4 of the chain's length remains ion the table, the center of mass is
(1/2)(3L/4) = 3L/8 below the table surface. The change in CM height is L/4. At that time, the velocity V is given by
M g L/4 = (1/2) M V^2
g L = 2 V^2
V = sqrt(gL/2) = 2.96 m/s
LEt total mass be m
force pulling: masshanging over *g
force puling: mg*x/1.79 where x is the amount hanging over.
well, then force pulling is linear with the length hanging over, so the average force between .25x and .75 x is at .5x
or avg force= mg/2
Vf^2=Vi^2+2ad
Vf^2=0+2(avgforce/m)(1/2 1.79)
Vf= sqrt (g*1.79/2)
check my thinking.
Vf= sqrt (2*
ignore the last line,old typo. I agree with drwls, it is the same argument.
To solve this problem, we can use the principle of conservation of mechanical energy. The initial potential energy of the hanging portion of the chain is converted into kinetic energy as the chain falls and accelerates.
Step 1: Determine the initial potential energy (PE1) of the hanging portion of the chain.
PE1 = mgh
Here, m is the total mass of the chain and h is the height of the hanging portion of the chain above the table. Since one-quarter of the length is hanging, the height is (3/4)L, where L is the total length of the chain.
Step 2: Determine the final potential energy (PE2) of the remaining portion of the chain on the table.
PE2 = mgh
Since only one-quarter of the length is remaining on the table, the height is (1/4)L.
Step 3: Calculate the change in potential energy (ΔPE) of the chain.
ΔPE = PE2 - PE1
Step 4: Calculate the change in kinetic energy (ΔKE) of the chain.
ΔKE = -ΔPE (by the principle of conservation of mechanical energy)
As the chain falls, the change in potential energy is converted into kinetic energy.
Step 5: Calculate the speed of the chain (v) at the given instant.
KE = (1/2)mv^2
Since ΔKE = -ΔPE, we can substitute this equation into the previous equation.
(1/2)mv^2 = -ΔPE
Solve for v.
Note: In this problem, assume that the chain remains straight and there is no friction between the chain and the table.