# Find how hollow a rock is given its density and its weight in water

• season89
In summary, by using the equation for apparent weight and the known densities of the rock and water, you can determine that the solid part of the rock makes up 80% of its volume.
season89

## Homework Statement

A geologist thinks a rock is hollow. The rock weighed in air is twice as much as when it's in water. Find the fraction of the rock's volume that is solid given the solid part of the rock is 2500kg/m^3 and water is 1000kg/m^3

## Homework Equations

rho = mass/volume
Fb = rho*volume*gravity
F(normal) = Force gravity (rock in air) - rho(fwater)*V(displaced)*gravity

## The Attempt at a Solution

Can you please show your attempt at the solution as per PF guidelines.

Jared

Given that density of solid part of rock is 2500kg/m^3, water is 1000kg/m^3, air has neglibile density and the whole object weighs twice is much in air than in water:

Apparent weight = m(object in air)*g - m(fluid displaced)*g
1/2*m(object in air)*g = m(object in air)*g - m(fluid displaced)g
1/2* rho (object)*V*g = rho (object)*V*g - rho (fluid)*V*g
1/2* rho (object) = rho (object) - rho (fluid)
-1/2*rho (object) = - 1000kg/m^3
rho (object) = 2000kg/m^3

therefore
2000kg/m^3 = x*(solid part) + (1-x)*(air)
2000kg/m^3 = 2500kg/m^3*x
x = 2000/2500 = 0.8 or 80%

To find the fraction of the rock's volume that is solid, we need to first calculate its volume and then compare it to the total volume of the rock. We can use the given density of the solid part of the rock (2500kg/m^3) and the weight of the rock in water to find its volume.

First, let's calculate the volume of the rock in air. We can use the formula rho = mass/volume to find the volume. Since the rock weighs twice as much in air as it does in water, we can say that the mass of the rock in air is twice the mass of the rock in water. Therefore, the volume of the rock in air is also twice the volume of the rock in water.

Next, we can use the formula Fb = rho*volume*gravity to find the buoyant force acting on the rock when it is submerged in water. Since the rock is in equilibrium (neither sinking nor floating), we know that the buoyant force is equal to the weight of the rock in water. Therefore, we can set Fb = weight of rock in water and solve for the volume of the rock in water.

Once we have the volume of the rock in water, we can use the formula F(normal) = Force gravity (rock in air) - rho(fwater)*V(displaced)*gravity to find the volume of water displaced by the rock. This volume of water displaced is equal to the volume of the hollow space inside the rock.

Finally, we can calculate the fraction of the rock's volume that is solid by dividing the volume of the solid part of the rock (2500kg/m^3) by the total volume of the rock (volume of solid part + volume of hollow space). This will give us the ratio of solid volume to total volume, and we can express it as a fraction or percentage.

In summary, to find the fraction of the rock's volume that is solid, we need to calculate the volume of the rock in air, the volume of water displaced by the rock, and then use these values to find the fraction of solid volume to total volume.

## 1. How do you determine the density of a rock?

The density of a rock can be determined by dividing its mass by its volume. This can be done by weighing the rock and then measuring its volume through displacement in a graduated cylinder.

## 2. What is the equation for finding the volume of a rock?

The volume of a rock can be found by dividing its weight in water by the difference between its weight in air and weight in water, and then multiplying by the density of water (1 g/cm^3).

## 3. How does the weight of a rock change when submerged in water?

The weight of a rock will appear to decrease when submerged in water due to the buoyant force acting on it. This is because the water exerts an upward force on the rock, which counteracts the force of gravity.

## 4. Can the density of a rock vary?

Yes, the density of a rock can vary depending on its composition and the presence of any voids or cracks within it. This is why it is important to take multiple measurements and calculate an average density.

## 5. How accurate is the method for finding the hollow space within a rock?

The accuracy of this method depends on the accuracy of the measurements taken and the assumptions made about the density of the rock. It is important to use precise instruments and take multiple measurements to improve accuracy.

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