A piece of wood of uniform cylindrical cross section floats upright in a liquid of relative density 0.80 with 1/6 of its height above the liquid. If placed in water, what is the fraction of its height above the water level?
volume of wood = V
density of wood = rhowood
density of liquid = .8 * rho water = 800 kg/m^3
density of water = 1000 kg/m^3
in liquid
rhowood * g * V = (5/6) * 800 *g * V
so
rhowood = 5 *800 / 6 = 2000/3 = 2/3 * density of water
so 1/3 of the wood is out of water
To find the fraction of the wood's height above the water level, we first need to determine the relative density of water.
Relative density (RD) is defined as the density of a substance divided by the density of a reference substance. In this case, the reference substance is water, so we can write:
RD = Density of wood / Density of water
Given that the wood floats upright with 1/6 of its height above the liquid, we can conclude that the fraction of the wood's volume below the liquid is 5/6. Since both the wood and liquid occupy the same volume, this implies that the fraction of the liquid's volume above the wood is also 5/6.
Now, let's solve the problem step by step:
Step 1: Determine the relative density of the liquid.
Given that the wood floats with 1/6 of its height above the liquid, we can equate this to the fraction of the liquid's volume above the wood:
5/6 = 1 - 1/6
5/6 = 5/6
So, the relative density of the liquid is 0.80.
Step 2: Determine the relative density of water using the formula for relative density:
RD = Density of wood / Density of water
0.80 = Density of wood / Density of water
Step 3: Find the fraction of the wood's height above the water level.
Since the relative density of water is equal to 1, we can rewrite the formula as:
1 = Density of wood / Density of water
By cross-multiplying, we get:
Density of wood = Density of water
This implies that the wood will float fully submerged in water, with no part above the water level. Therefore, the fraction of the wood's height above the water level is 0.
In conclusion, when placed in water, none of the wood's height will be above the water level.
To determine the fraction of the wood's height above the water level, we need to understand the concept of buoyancy and how it applies to floating objects.
Buoyancy is the upward force exerted by a fluid (in this case, a liquid) on an object immersed or floating in it. This force is equal to the weight of the fluid displaced by the object.
When an object floats, it means that the buoyant force acting on the object is equal to its weight. This equilibrium can be expressed by Archimedes' principle:
Buoyant force = Weight of the object
In this case, the piece of wood floats upright in a liquid with a relative density of 0.80. Relative density is the ratio of the density of a substance to the density of a reference substance. For liquids, the reference substance is usually water.
Given that 1/6 of the height of the wood is above the liquid in which it floats, we can assume that 5/6 of its height is submerged in the liquid. This implies that the buoyant force acting on the wood is providing enough upward force to counterbalance 5/6 of its weight.
Now, the relative density of the liquid is 0.80, which means its density is 0.80 times that of water. We can use this information to determine the fraction of the wood's height above the water level.
To start, we need to understand the relationship between the relative densities and the fractions of an object's height above and below the liquid level.
The fraction of an object's height submerged in a liquid is equal to:
1 - (Relative Density of the liquid / Relative Density of the object)
Let's substitute the values we have:
Fraction submerged in the liquid = 1 - (0.80 / 1) = 1 - 0.80 = 0.20
This implies that 0.20 (or 1/5) of the wood's height is submerged in the liquid, while 4/5 of its height remains above the liquid.
Now, since we are interested in determining the fraction of the wood's height above the water level, we need to consider the density of water. Given that the density of water is 1 g/cm³, the relative density of water is 1/1 = 1.
Using the same formula as before, we can now calculate the fraction of the wood's height above the water level:
Fraction above water level = 1 - (Relative Density of water / Relative Density of the object)
Fraction above water level = 1 - (1 / 1) = 1 - 1 = 0
Therefore, the fraction of the wood's height above the water level is 0, meaning that none of its height is above the water level when placed in water.