A flat ice floe, the base area of which S = 4.0 m2 is able to hold a person weighing m = 60 kg

A flat ice floe, the base area of which S = 4.0 m2 is able to hold a person weighing m = 60 kg on the water. Determine the thickness of the ice floe.

S = 4 m2.

mh = 60 kg.

g = 10 N / kg.

ρw = 1000 kg / m3.

ρl = 900 kg / m3.

d -?

Two forces act on an ice floe with a person in the water: the force of gravity Ft, directed vertically downward, and the buoyancy force of Archimedes Farkh, directed vertically upward.

Ft = Farch.

The force of gravity is determined by the formula: Ft = m * g, where m is the mass of the ice floe with a person, g is the acceleration of gravity.

m = ml + mh.

The buoyancy force of Archimedes Farch is determined by the formula: Farch = ρw * g * V. Where ρ is the density of the liquid in which the body is immersed, g is the acceleration of gravity, V is the volume of the ice floe.

Let us express the volume of the ice floe by the formula: V = S * d, where d is the thickness of the ice floe.

ml = V * ρl = S * d * ρl.

(S * d * ρl + mh) * g = ρw * g * S * d.

S * d * ρl * g + mh * g = ρw * g * S * d.

ρw * g * S * d – S * d * ρl * g = mh * g.

ρw * S * d – S * d * ρl = mh.

The thickness of the ice floe d will be determined by the formula: d = mh / (ρw – ρl) * S.

d = 60 kg / (1000 kg / m3 – 900 kg / m3) * 4 m2 = 0.15 m.

Answer: the ice floe has a thickness of d = 0.15 m.