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

[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

 

[JaredJames]

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

Jared

 

[season89]

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%

 

[rl.bhat]

Your answer is correct.

 

[rwisz]

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.