An original hydrometer of mass 28g
Floats with 6cm of its stem out of water. The area or cross-section of the stemost is 0.75cm^2. Find the total volume of hydrometer and length of the stem above the surface when it floats in a liquid of relative density 1.4
60cm^3 and 8cm
Ah, the mighty hydrometer! Let's dive into this buoyant adventure, shall we?
To find the total volume of the hydrometer, we need to consider the volume of the stem above the surface and the volume of the submerged portion.
First, let's calculate the volume of the submerged portion. Given that the hydrometer floats with 6cm of its stem out of water, we can say that the submerged length is the total length minus the length above the surface. So, the submerged length is 6 cm.
The area or cross-section of the stem is given as 0.75 cm^2. To find the volume of the submerged portion, we'll multiply the submerged length by the cross-sectional area:
Volume_submerged = submerged length * cross-sectional area
= 6 cm * 0.75 cm^2
Now, let's calculate the volume of the entire hydrometer by considering the density of the liquid it floats in. The relative density of a liquid is the ratio of its density to the density of water, and we are given that the relative density is 1.4.
Volume_total = Volume_submerged / Relative_density
= (6 cm * 0.75 cm^2) / 1.4
And there you have it! You now have the total volume of the hydrometer. As for the length of the stem above the surface, you already know that it's 6 cm.
Just remember, my dear friend, that this answer assumes that the hydrometer is a perfect cylinder, with a uniform cross-section and no weirdly shaped fluff at the top (like a clown nose!). So, take it with a pinch of clownish wisdom.
To find the total volume of the hydrometer, we need to determine the volume of the part submerged in water and the volume of the stem above the surface.
1. Volume submerged in water:
When the hydrometer floats, it displaces its own weight in water. The buoyancy force acting on the hydrometer is equal to the weight of the water displaced.
Since the mass of the hydrometer is 28g and the density of water is 1g/cm^3, the weight of the hydrometer is 28g. This means the volume of water displaced is also 28cm^3.
2. Volume of the stem above the surface:
To find the volume of the stem above the surface, we need to determine the length of the stem that is above the liquid's surface.
Given that the hydrometer initially floats with 6cm of its stem out of water, when it floats in a liquid of relative density 1.4, we can use the principle of flotation to determine the length of the stem above the surface.
The principle of flotation states that the weight of the displaced liquid is equal to the weight of the object.
Let L be the length of the stem above the surface.
The volume of the stem above the surface can be calculated using the formula: Volume = Area × Length
We know the area of the stem = 0.75 cm^2 and the relative density of the liquid = 1.4.
So, the weight of the hydrometer is 28g × 1.4 = 39.2g.
The weight of the liquid displaced is the weight of the hydrometer above the surface = 0.75 × L cm^3 × 39.2g/cm^3.
We can set up the equation: 0.75 × L × 39.2 = 0.75 × 6 × 1.
Now, we can solve for L:
L = (0.75 × 6 × 1) / (0.75 × 39.2)
= 0.115 cm
So, the length of the stem above the surface is approximately 0.115 cm.
3. Total volume of the hydrometer:
The total volume of the hydrometer is the sum of the volume submerged in water and the volume of the stem above the surface.
Total volume = Volume submerged in water + Volume of stem above the surface
= 28 cm^3 + 0.115 cm^3
≈ 28.115 cm^3
Therefore, the total volume of the hydrometer is approximately 28.115 cm^3 and the length of the stem above the surface when it floats in a liquid of relative density 1.4 is approximately 0.115 cm.
To find the total volume of the hydrometer and the length of the stem above the surface when it floats in a liquid of relative density 1.4, we need to use the principle of flotation.
The principle of flotation states that when an object is floating in a liquid, the weight of the liquid displaced (buoyant force) is equal to the weight of the object.
First, let's find the volume of the hydrometer. We can do this by dividing its mass by the density of water.
Density of water = 1 g/cm^3
Volume of hydrometer = Mass of hydrometer / Density of water
Volume of hydrometer = 28 g / 1 g/cm^3
Volume of hydrometer = 28 cm^3
Next, let's find the volume of the submerged portion of the hydrometer. We can do this by multiplying the length of the submerged stem by the cross-sectional area of the stem.
Length of submerged stem = Total length of stem - Length of stem above the surface
Length of submerged stem = 6 cm - Length of stem above the surface
The volume of the submerged portion of the hydrometer is then given by:
Volume of submerged portion = Length of submerged stem * Area of stem
Volume of submerged portion = (6 cm - Length of stem above the surface) * 0.75 cm^2
Since the hydrometer is floating, the weight of the liquid displaced is equal to the weight of the hydrometer. We can express this as:
Weight of liquid displaced = Weight of hydrometer
The weight of the liquid displaced is given by:
Weight of liquid displaced = Density of liquid * Volume of submerged portion * g
The weight of the hydrometer is given by:
Weight of hydrometer = Mass of hydrometer * g
Since the buoyant force is equal to the weight of the liquid displaced:
Buoyant force = Weight of liquid displaced
Density of liquid * Volume of submerged portion * g = Mass of hydrometer * g
We can cancel out the g factors:
Density of liquid * Volume of submerged portion = Mass of hydrometer
Since the relative density of the liquid is given as 1.4, we can write:
Density of liquid = 1.4 * Density of water
Substituting these values:
1.4 * Density of water * Volume of submerged portion = Mass of hydrometer
Now we can solve for the length of the submerged stem and the volume of the hydrometer.
1.4 * 1 g/cm^3 * (6 cm - Length of stem above the surface) * 0.75 cm^2 = 28 g
Simplifying the equation gives:
1.05 * (6 cm - Length of stem above the surface) = 28/1.4
Simplifying further:
6 cm - Length of stem above the surface = 20 cm
Rearranging the equation:
Length of stem above the surface = 6 cm - 20 cm
Simplifying:
Length of stem above the surface = -14 cm
Since a negative length doesn't make sense in this context, we can conclude that there has been an error in the calculations.