a cube of wood floating in water support 1 kg mass resting on the center of its top surface. when the mass is removed the cube rise 2.5cm find the length of
cube? ans is 20cm
To find the length of the cube, we can use the concept of buoyancy and the principle of equilibrium.
Let's break down the problem step by step:
Step 1: Identify the given information:
- The cube of wood floats in water.
- The cube supports a 1 kg mass when it is placed on its top surface.
- When the 1 kg mass is removed, the cube rises 2.5 cm.
Step 2: Understand the concept of buoyancy:
When an object is submerged in a fluid, such as water, it experiences a buoyant force. This force is equal to the weight of the fluid displaced by the object. For a floating object, the weight of the object is balanced by the buoyant force.
Step 3: Determine the weight of the cube when it supports the 1 kg mass:
Since the cube is floating, the weight of the cube must be equal to the weight of the 1 kg mass. We know that the weight of the 1 kg mass is equal to its mass multiplied by the acceleration due to gravity (9.8 m/s^2). Therefore, the weight of the cube is also 1 kg × 9.8 m/s^2 = 9.8 N.
Step 4: Determine the volume of the cube:
The buoyant force on the cube is equal to the weight of the water displaced by the cube. Since the cube is completely submerged when the mass is placed on it, the volume of water displaced is equal to the volume of the cube. We need to find the length of the cube, so let's denote the length as "L".
The volume of a cube is given by V = L^3.
Step 5: Calculate the buoyant force:
The buoyant force on the cube is equal to the weight of the cube, which is 9.8 N.
Step 6: Determine the weight of the water displaced:
The weight of the water displaced is equal to the buoyant force, which is 9.8 N.
Step 7: Determine the volume of the water displaced:
We can use Archimedes' principle, which states that the weight of the fluid displaced is equal to the weight of the object immersed in the fluid.
The weight of the water displaced is equal to the weight of the cube, which is 9.8 N. Since the density of water is 1000 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2, we can use the formula:
Weight of water displaced = Density × Volume × g
Simplifying the equation, we have:
9.8 N = 1000 kg/m^3 × Volume × 9.8 m/s^2
Step 8: Solve for the volume of the cube (Volume):
Since the volume of the cube is the same as the volume of the water displaced, we can rearrange the equation from the previous step to solve for the volume of the cube:
Volume = 9.8 N / (1000 kg/m^3 × 9.8 m/s^2) = 0.001 m^3
Step 9: Solve for the length of the cube (L):
Using the formula for the volume of a cube (V = L^3), we can substitute the known volume (0.001 m^3) into the equation and solve for the length (L):
L^3 = 0.001 m^3
Taking the cube root of both sides, we have:
L = ∛(0.001 m^3) = 0.1 m = 10 cm
Therefore, the length of the cube is 10 cm, not 20 cm as mentioned in the answer.
mass=massofwaterdisplaced=1g/cm^3(2.5*L^2
1000=1*2.5*L^2
L=sqrt400=20cm