Question

C A M C 11. AU shape vessel shown consists of two cylindrical vertical arms A and B of cross-sectional area Sa = 300 cm and Sp = 500 cm. A very light piston can slide without friction in the arm A and confines water below it. A thin light thread attached to the piston at one of its ends, passes under two ideal pulleys fixed at the bottom of the vessel and then attached at the other end to the bottom of a very light cube C of side 1 = 10 cm. Initially, when a block of mass m is placed on the piston, the cube floats with negligible portion inside the water and the thread is not slack. How high will the piston move, if the block is gradually lifted off the piston? Density of water is po = 1000 kg/m3.

Please provide correct solution this time.

Answer 1

11. For the given U shaped vessel
Given, Sa = 300 cm^2
Sb = 500 cm^2
the cube is very light, and so are pistons

side of cube, l = 10 cm
when m is on the piston, C floats on the other side
so height difference between two surfaces is h
rho*g*h = mg/Sa
h = mg/(Sa*rho*g)
rho = 1000 kg/m^3
g = 9.81 m/s/s
h = m/(1000*300*10^-4)
h = m/30

now, when m is removed, the water level in both the arms should not be the same becuase of the force of buoyancy on C, which is transmitted to the piston which keeps the forces on the water as such that there is a level difference
let this final difference be h'

extra length of the string needed to connect the two surfaces through water at level h' is s
let say water in right arm went down by distance x and in left arm got up by distance y
then
x + y + h' = h
also
s = -x + y
also,
x*Sb = y*Sa
so, y = x(Sb/Sa) = x(500/300) = 5x/3
hence
s = -x + 5x/3 = 2x/3
x + 5x/3 + h' = h
hence
8x/3 + h' = m/30

now, this extra length required will cause the block C to get submerged
so, buoyant force is B = rho*l^2*s*g
hence
rho*g*h' = rho*l^2*s*g /Sa
h' = l^2*s /Sa

8x/3 + l^2*s /Sa = m/30
8x/3 + (0.1^2)*(2x/3)/(300*10^-4) = m/30
2.888888888*x = m/30
x = 0.01153846153*m = m/86.666666666

hence the piston will move up by distance x = m/86.6666666 m
where m is mass of the block in kg