Question

The volume of 50g of a substance is 20 $c{{m}^{3}}$. If the density of water is 1$gc{{m}^{-3}}$, find the density of the substance and interpret whether the substance will sink or float?
$\text{A}\text{. }2.5gc{{m}^{-3}}\text{, substance will sink}$
$\text{B}\text{. }2.5gc{{m}^{-3}}\text{, substance will float}$
$\text{C}\text{. 1}.5gc{{m}^{-3}}\text{, substance will neither sink nor float}$
$\text{D}\text{. 1}.5gc{{m}^{-3}}\text{, substance will float inside water }$

Answer 1

Hint: Density of substance is the amount mass present in one unit of volume of the substance. The formula for density of a substance is $\dfrac{\text{mass}}{\text{volume occupied by the mass}}$.
When a substance is put into a liquid an upward force is exerted on the liquid due to which the substance may float.

Formula used:
$\text{density}=\dfrac{\text{mass}}{\text{volume occupied by the mass}}$
${{F}_{B\max }}=\rho gV$
${{F}_{g}}=mg$

Complete step by step answer:
Density of substance is the amount of mass present in one unit of volume of the substance. The formula for density of a substance is $\dfrac{\text{mass}}{\text{volume occupied by the mass}}$ .

Hence, the density of the given substance of mass of 50 g is $\dfrac{50}{20}=2.5gc{{m}^{-1}}$.

When a substance is put into a liquid, the liquid exerts an upward force on the substance. This upward force exerted on the substance, by the liquid is called buoyant force. Due to this force, the substance in the liquid may float on the liquid.

When the substance is in the liquid (fully or partially), there are two forces acting on the substance. One is the gravitational force (mg) exerted by earth and the other is the buoyant force exerted by the liquid.

The gravitational force acts downwards and the buoyant force acts upwards. If the buoyant force overcomes the gravitational force, the substance will float. Otherwise, the net force will act downwards and therefore the substance will sink.

Now, the maximum value of buoyant force is equal to the product of the volume of the substance (V), the density of the liquid ($\rho$) and g (acceleration due to gravity).
i.e. ${{F}_{B\max }}=\rho gV$

Let us calculate the value of ${{F}_{B\max }}$ for the given substance.
Here, $\rho =1gc{{m}^{-3}}$ and V = 20 $c{{m}^{3}}$.

Therefore, ${{F}_{B\max }}=\rho gV=(1)(20)g=(20g)gcm{{s}^{-2}}$.

The value of gravitational force will be equal to ${{F}_{g}}=mg=(50g)gcm{{s}^{-2}}$.

As you can see the maximum value of buoyant force is less than the gravitational force.
Therefore, the substance will sink into water.

Hence, the correct option is A.

Note: We know that $\text{density = }\dfrac{\text{mass}}{\text{volume}}$.
Let the density of the given substance be $\delta$,

Therefore, $\delta =\dfrac{m}{V}\Rightarrow m=\delta V$.

Therefore, the gravitational force can be written as ${{F}_{g}}=mg=\delta gV$ …….. (i).
And the maximum buoyant force is equal to ${{F}_{B\max }}=\rho gV$ ……. (ii).

Compare (i) and (ii),

If the density of the liquid is more than the density of the substance then the maximum buoyant force will be greater than gravitational force.

Hence, the substance will float.

If the density of the liquid is less than the density of the substance then the maximum buoyant force will be less than gravitational force.

Hence, the substance will sink.

Therefore, we can solve this type of question by just comparing the densities of the liquid and the substance.