Question

A cubical solid aluminium (bulk modulus = -V$\dfrac{dp}{dv}$ = 70 GPa) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be 10³ kg m^−3 and 10 ms^−2, respectively, the change in the edge length of the block in mm is _____.

Answer 1

it has given that, bulk modulus of a solid aluminium block is 70 GPa, edge length of block is 1 m placed on the earth's surface. it is kept on the floor of a 5km deep ocean.

To find : the change in edge length of the block on mm is, ...

solution : let a is edge length of cubic block

then volume of block , V = a³

differentiating both sides we get,

dV = 3a²

⇒dV/V = 3da/a ....(1)

now, bulk modulus of block, B = - V dP/dV

here putting, P = ρgh and dV/V = 3a/a [ from eq (1)

B = -ρgha/3da

now putting, B = 70 × 10^9 Pa

ρ = 10³ kg/m³ , g = 10 m/s² , h = 5000 m, a = 1 m

so, 70 × 10^9 = -(10³ × 10 × 5000 × 1)/(3da)

⇒da = (5 × 10^7)/(3 × 70 × 10^9) = 5/21 × 10^-3

= 0.238 ≈ 0.24 mm

Therefore the change in edge length of the block is in mm is 0.24