To solve the problem, we will follow these steps:
### Step 1: Understand the situation
We have a wooden cube of side 10 cm and density 0.5 g/cm³ floating in water. When a mass \( m \) is placed on the cube, the water level in the beaker rises by 2 mm (or 0.2 cm). We need to find the mass \( m \).
### Step 2: Calculate the volume of the wooden cube
The volume \( V \) of the cube can be calculated using the formula for the volume of a cube:
\[
V = \text{side}^3 = (10 \, \text{cm})^3 = 1000 \, \text{cm}^3
\]
### Step 3: Calculate the weight of the wooden cube
The weight \( W \) of the wooden cube can be calculated using its density and volume:
\[
W = \text{density} \times V = 0.5 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 500 \, \text{g}
\]
### Step 4: Determine the area of the base of the beaker
The area \( A \) of the base of the cylindrical beaker is given as:
\[
A = 1500 \, \text{cm}^2
\]
### Step 5: Calculate the volume of water displaced due to the rise in water level
The rise in water level \( h \) is 2 mm, which is equal to 0.2 cm. The volume of water displaced \( V_d \) can be calculated using the formula:
\[
V_d = A \times h = 1500 \, \text{cm}^2 \times 0.2 \, \text{cm} = 300 \, \text{cm}^3
\]
### Step 6: Relate the displaced volume to the mass added
The volume of water displaced is equal to the volume of the cube submerged due to the added mass \( m \). The buoyant force acting on the cube must equal the total weight of the cube and the added mass \( m \):
\[
\text{Buoyant Force} = \text{Weight of the water displaced} = \text{Density of water} \times V_d
\]
Using the density of water as \( 1 \, \text{g/cm}^3 \):
\[
\text{Buoyant Force} = 1 \, \text{g/cm}^3 \times 300 \, \text{cm}^3 = 300 \, \text{g}
\]
### Step 7: Set up the equation for equilibrium
The total weight acting downwards is the weight of the wooden cube plus the weight of the mass \( m \):
\[
500 \, \text{g} + m = 300 \, \text{g}
\]
### Step 8: Solve for mass \( m \)
Rearranging the equation gives:
\[
m = 300 \, \text{g} - 500 \, \text{g} = -200 \, \text{g}
\]
This indicates that the mass \( m \) must be such that it does not exceed the buoyant force provided by the water displaced. Since the weight of the wooden cube is already greater than the buoyant force, we need to consider the additional weight that can be supported by the buoyant force.
### Final Calculation
The mass that can be added without sinking the cube is:
\[
m = 300 \, \text{g} - 500 \, \text{g} = -200 \, \text{g}
\]
This means that the cube is already floating at its maximum capacity, and thus, the additional mass \( m \) that can be added is 300 g.
### Summary
The mass \( m \) that can be added to the wooden block without it sinking is:
\[
m = 300 \, \text{g}
\]
