Question

Открытая сверху коробка в форме куба плавает так, что в воде оказывается её половина. Внутрь коробки аккуратно кладут груз массой m. При какой минимальной массе груза коробка утонет? Масса коробки M = 400г

Answer 1

Я решил на прикрепленном листочке.

Answer 2

Problem Analysis

We have an open-top box in the shape of a cube floating in water such that half of the box is submerged. The mass of the box is given as 400g, and we need to determine the minimum mass of the load that will cause the box to sink.

Solution

To solve this problem, we need to consider the buoyant force acting on the box and the load. The buoyant force is equal to the weight of the water displaced by the submerged portion of the box and the load. If the buoyant force is greater than or equal to the combined weight of the box and the load, the box will float. If the buoyant force is less than the combined weight, the box will sink.

Let's denote the mass of the load as m and the mass of the box as M. The buoyant force can be calculated using Archimedes' principle:

Buoyant force = Weight of water displaced = Density of water x Volume of water displaced x g

Since the box is in the shape of a cube, the volume of water displaced is equal to half the volume of the box. The density of water is approximately 1000 kg/m^3, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

The combined weight of the box and the load is given by:

Weight = (m + M) x g

To determine the minimum mass of the load that will cause the box to sink, we need to find the mass m such that the buoyant force is less than the weight of the box and the load.

Let's calculate the minimum mass of the load using the given information.

Calculation

Given: - Mass of the box (M) = 400g = 0.4kg - Density of water = 1000 kg/m^3 - Acceleration due to gravity (g) = 9.8 m/s^2

We need to find the minimum mass of the load (m) that will cause the box to sink.

To calculate the minimum mass of the load, we can equate the buoyant force to the weight of the box and the load:

Buoyant force = Weight

Using Archimedes' principle, the buoyant force can be calculated as:

Buoyant force = Density of water x Volume of water displaced x g

The volume of water displaced is equal to half the volume of the box, which is equal to half the volume of a cube with side length s.

Let's calculate the side length of the cube:

s = (Volume of the box)^(1/3)

The volume of the box is given by:

Volume of the box = s^3

Substituting this into the equation for the buoyant force:

Buoyant force = Density of water x (0.5 x s^3) x g

Now, let's equate the buoyant force to the weight of the box and the load:

Density of water x (0.5 x s^3) x g = (m + M) x g

Simplifying the equation:

Density of water x (0.5 x s^3) = m + M

Substituting the values:

1000 kg/m^3 x (0.5 x s^3) = m + 0.4 kg

Simplifying further:

500 x s^3 = m + 0.4

Now, we can solve for the minimum mass of the load (m) by substituting the value of s:

500 x (Volume of the box)^(1/3) = m + 0.4

The volume of the box is given by:

Volume of the box = s^3

Substituting this into the equation:

500 x (s^3)^(1/3) = m + 0.4

Simplifying:

500 x s = m + 0.4

Now, we can substitute the value of s into the equation:

500 x (Volume of the box)^(1/3) = m + 0.4

500 x (s^3)^(1/3) = m + 0.4

500 x s = m + 0.4

Substituting the value of s:

500 x (0.5 x s^3) = m + 0.4

250 x s^3 = m + 0.4

Now, we can solve for the minimum mass of the load (m):

m = 250 x s^3 - 0.4

Let's calculate the value of m.

Calculation Continued

To calculate the minimum mass of the load (m), we need to find the value of s.

The volume of the box is given by:

Volume of the box = s^3

Substituting the value of the mass of the box (M) into the equation:

0.4 kg = s^3

Taking the cube root of both sides:

s = (0.4 kg)^(1/3)

Calculating the value of s:

s ≈ 0.736

Now, we can substitute the value of s into the equation for m:

m = 250 x (0.736)^3 - 0.4

Calculating the value of m:

m ≈ 125.81 g

Therefore, the minimum mass of the load that will cause the box to sink is approximately 125.81 grams.

Answer

The minimum mass of the load that will cause the box to sink is approximately 125.81 grams.