Question

A cylindrical block of area of cross-section $A$ and of material of density $\rho$ is placed in a liquid one-third of density of block. The block compresses a spring and compression in the spring is one-third of the length of the block. If acceleration due to gravity is $g$, the spring constant of the spring is:

• A. $\rho Ag$
• B. $2\rho Ag$
• C. $2\rho Ag/3$
• D. $\rho Ag/3$

Answer 1

Answer: (B)

$2\rho Ag$

Since the block is at rest, net force on the cylindrical block is zero, i.e. net upward =net downward force
$F_s+F_b=W$
where
$F_s=$ force due to compression in the spring
$F_b=$buoyant force
$W=$ weight of the cylindrical block
$\Rightarrow kx+AL\frac{\rho }{3}g=AL\rho g\Rightarrow k\frac{L}{3}+AL\frac{\rho }{3}g=AL\rho g\Rightarrow k\frac{L}{3}=\frac{2AL\rho g}{3}\Rightarrow k=2\rho Ag$