buoyancy, Buoyant Force

Question

At a party, 3 engineers (MP = 70 kg each) ponder whether a Jacuzzi with a capacity of 1500x1500x500 mm3 (length L x width W x height H) and a dead weight of MJ = 250 kg is sufficient to power a 1964 Mini ( car) with a weight of MM = 650 kg (without passengers) safely floating on a river. The density of river water is ρW ater = 1.05 g cm3 . Of course, the calculation has to be realistic, which is why the swell of the river of W = 27 cm is taken into account. Is the capacity of the air-filled jacuzzi sufficient to allow the mini, which is occupied by all 3 engineers, to float safely on the river? The jacuzzi should not fill up with water from the waves!

Answer 1

Let

g = 9.81 m/s² be gravitational acceleration,

S = 150 cm x 150 cm = 22500 cm² be the cross section of the jacuzzi,

h = 50 cm - 27 cm = 23 cm be the maximum safe depth to which the jacuzzi can be submerged.

When the jacuzzi is submerged to the maximal depth, it displaces water of volume

Sh

This water has weight

Shρg

By the law of Archimedes, that is the maximum weight that the submerged jacuzzi can possibly support.

The total mass that the jacuzzi is to support is

3MP + MJ + MM

That has has the weight

(3MP + MJ + MM)g

The jacuzzi can support that weight if

(3MP + MJ + MM)g ≤ Shρg

Or equivalently

3MP + MJ + MM ≤ Shρ

Substituting actual numbers

3×70 kg + 250 kg + 650 kg ≤ 22500 cm² × 23 cm × 1.05 g/cm³

1110 kg ≤ 543375 g = 543.375 kg

Since the inequality is not satisfied the jacuzzi cannot be used safely to float the car.