Question

Given a block of wood with a height of 10.0 cm. When placed in water, 9.00 cm of this block's height are above water. What is the density of this block in units of grams/cm^3? A)1 B)10 C)0.9 D)0.1 E)9

Answer 1

We are given a wooden block that floats in water. The given problem can be exemplified in the following diagram:

To determine the density of the block we need to use the fact that, according to Archimedes principle, the weight of the displaced liquid is equal to the weight of the object, therefore, we have:

`w_{\text{water}}=w_{\text{block}}`

Where:

`\begin{gathered} w_{\text{water}}=\text{ weight of water} \\ w_{\text{block}}=\text{ weight of the block} \end{gathered}`

The weight is the product of the mass and the acceleration of gravity, therefore, we have:

`m_{\text{water}}g=m_{\text{block}}g`
`\begin{gathered} m=\text{ mass} \\ g=\text{ acceleration of gravity} \end{gathered}`

Now, the mass is the product of the density and the volume, therefore, we have:

`\rho_(water)m_(water)g=\rho_(block)m_(block)g`
`\rho=\text{ density}`

We can cancel out the acceleration of gravity and we get:

`\rho_(water)V_(water)=\rho_(block)V_(block)`

Now, the volume of the water is equivalent to the height that is underwater multiplied by the width and the length of the block:

`\rho_(water)(1cm)(w)(l)=\rho_(block)V_(block)`

Where:

`\begin{gathered} w=\text{ width} \\ l=\text{ length} \end{gathered}`

Also, the volume of the block is the height of the block multiplied by the width and the height:

`\rho_(water)(1cm)(w)(l)=\rho_(block)(10cm)(w)(l)`

Now, we cancel out the width and the length

`\rho_(water)(1cm)=\rho_(block)(10cm)`

Now, we solve for the density of the block by dividing both sides by 10:

`(\rho_(water)(1cm))/(10cm)=\rho_(block)`

The density of water in g/cm^3 is given by:

`\rho_(water)=1(g)/(cm^3)`

Substituting we get:

`((1(g)/(cm^3))(1cm))/(10cm)=\rho_(block)`

Solving we get:

`(1)/(10)(g)/(cm^3)=\rho_(block)`

In decimal form this is:

`0.1(g)/(cm^3)=\rho_(block)`

Therefore, the density of the block is 0.1 g/cm^3