Figure 3.50 shows a wooden cube floating in water. When a stone is placed on the wood, it sinks a further 0.2m.
Find the weight of the
(a) wood
(b) stone
(Density of water = 1000kgm^-3]​

(a) The weight of the wood is approximately 5,886 N

(b) The weight of the stone is approximately 1,962 N

Explanation:

The given parameters are;

The base area of the wood, A = 1 m × 1 m = 1 m²

The depth to which the wood is submerged below the water, d = 0.6 m

The depth further the wood sinks when a stone is placed on the wood, Δd = 0.2 m

The density of water, ρ = 1,000 kg/m³

(a) Archimedes' principle states that the up-thrust, or upward force acting on an object submerged in a fluid, is equal to the weight of the displaced fluid.

For a floating object such as the wood, the up-thrust is equal to the weight of the object

The weight of the water displaced, W = The volume of the displaced water, V × The density of the water, ρ × The acceleration due to gravity, g

W = V × ρ × g

'V' is the same as the volume of the submerged wood, therefore;

V = A × d = 1 m × 1 m × 0.6 m = 0.6 m³

g ≈ 9.81 m/s²

Therefore;

W = 0.6 m³ × 1,000 kg/m³ × 9.81 m/s² ≈ 5,886 Newtons

The weight of the water displaced, W ≈ 5,886 N = The up-thrust = The weight of the wood

∴ The weight of the wood, W ≈ 5,886 N

(b) Similarly, the weight of the stone, is equal to the weight of the additional volume of water displaced

∴ = A × Δd × ρ × g

By plugging in the values, we get

≈ 1 m² × 0.2 m × 1,000 kg/m³ × 9.81 m/s² = 1,962 N

The weight of the stone, ≈ 1,962 Newtons

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.