To find out how much of the rod is underwater, we can use the concept of buoyancy. Buoyancy is the upward force exerted by a fluid (in this case, water) on an object submerged in it.
First, let's calculate the volume of the rod submerged in water. We know that the entire length of the rod is 10 cm, and we want to find the portion submerged. Let's assume that 'x' cm of the rod is underwater.
The volume of the submerged portion can be calculated by multiplying the length of the submerged section by its cross-sectional area. Since the cross-sectional area is constant throughout the entire rod, we can simply calculate the volume by multiplying the cross-sectional area by the submerged length.
The volume of the submerged portion = Area of cross-section * Length submerged
Now, the cross-sectional area of the rod is the same as the area of the circular end. Therefore, the cross-sectional area can be calculated using the formula for the area of a circle, which is given by:
Area of a circle = π * radius^2
Since the radius of the circular end of the rod is 1 cm (since only 1 cm is made of metal), the cross-sectional area can be calculated as:
Area of cross-section = π * (1 cm)^2 = π cm^2
So, the volume of the submerged portion is given by:
Volume of submerged portion = π cm^2 * Length submerged
Next, we need to calculate the weight of the submerged portion. Since the metal part points downward, we have to consider the weight of only the metal part.
The weight of an object can be calculated using the formula:
Weight = Mass * gravity
The mass of the submerged portion can be calculated using its volume and density. The density of metal is given as 5000 kg/m^3, but we need to convert it to kg/cm^3 to match the units of the dimensions of the rod. One cubic meter is equal to 1000000 cubic centimeters, so we can convert the density as follows:
Density of metal = 5000 kg/m^3 = 5000 kg / 1000000 cm^3 = 0.005 kg/cm^3
Now, we can calculate the mass of the submerged portion of the rod as:
Mass of submerged portion = Density of metal * Volume of submerged portion
Finally, we need to calculate the weight of the submerged portion. Since the metal part points downward, the weight is given by:
Weight of submerged portion = Mass of submerged portion * gravity
Substituting the values into the equation, we get:
Weight of submerged portion = (Density of metal * Volume of submerged portion) * gravity
To determine how much of the rod is underwater, we need to find the balance point where the weight of the submerged portion is equal to the weight of the water displaced by the submerged portion.
When the rod is in equilibrium:
Weight of submerged portion = Weight of water displaced
To find the weight of the water displaced, we use the formula:
Weight of water displaced = Density of water * Volume of water displaced * gravity
Since the volume of water displaced is the same as the volume of the submerged portion, we can rewrite the equation as:
Weight of submerged portion = Density of water * Volume of submerged portion * gravity
Now, we can set the two equations equal to each other and solve for the Length submerged:
(Density of metal * Volume of submerged portion) * gravity = Density of water * Volume of submerged portion * gravity
The gravitational force (gravity) is the same on both sides, so we can cancel it out:
Density of metal * Volume of submerged portion = Density of water * Volume of submerged portion
Volume of submerged portion cancels out:
Density of metal = Density of water
Therefore, the length submerged is equal to the ratio of the density of metal to the density of water:
Length submerged = Density of metal / Density of water
Now we substitute the given densities:
Length submerged = 0.005 kg/cm^3 / 1 kg/cm^3
Simplifying the expression, we find:
Length submerged = 0.005
Therefore, the length submerged is 0.005 cm, which is equivalent to 0.05 mm.
So, only 0.05 mm of the rod is underwater.