Have **solved density exercises** It will help to better understand this term and to understand all the implications that density has when analyzing different objects.

Density is a term widely used in physics and chemistry, and refers to the relationship between the mass of a body and the volume it occupies.

Density is usually denoted by the Greek letter “ρ” (ro) and is defined as the ratio between the mass of a body and its volume. That is, the unit of weight is located in the numerator and the unit of volume in the denominator.

Therefore, the unit of measurement used for this scalar quantity is kilograms per cubic meter (kg/m³), but it can also be found in certain literature as grams per cubic centimeter (g/cm³).

**definition of density**

Previously it was said that the density of an object, denoted by “ρ” (ro) is the quotient between its mass “m” and the volume that it occupies “V”.

That is: ρ= m / V.

One consequence that follows from this definition is that two objects can have the same weight, but if they have different volumes, then they will have different densities.

In the same way it is concluded that two objects can have the same volume but, if their weights are different, then their densities will be different.

A very clear example of this conclusion is taking two cylindrical objects with the same volume, but one object is made of cork and the other is made of lead. The difference between the weights of the objects will cause their densities to be different.

**Solved problems of density**

**First exercise**

Raquel works in a laboratory calculating the density of certain objects. José brought Raquel an object whose weight is 330 grams and her capacity is 900 cubic centimeters. What is the density of the object that José gave to Raquel?

As said before, the unit of measure for density can also be g/cm³. Therefore, there is no need to convert units. Applying the previous definition, the density of the object that José brought to Raquel is:

ρ= 330g / 900 cm³ = 11g / 30cm³=11/30 g/cm³.

**second exercise**

Rodolfo and Alberto each have a cylinder and want to know which cylinder has the greater density.

Rodolfo’s cylinder weighs 500 g and has a volume of 1000 cm³ while Alberto’s cylinder weighs 1000 g and has a volume of 2000 cm³. Which cylinder has the greater density?

Let ρ1 be the density of Rodolfo’s cylinder and ρ2 the density of Alberto’s cylinder. By using the formula for the calculation of density, we obtain:

ρ1= 500/1000 g/cm³ = 1/2 g/cm³ and ρ2= 1000/2000 g/cm³ = 1/2 g/cm³.

Therefore, both cylinders have the same density. It should be noted that according to the volume and weight, it can be concluded that Alberto’s cylinder is larger and heavier than Rodolfo’s. However, their densities are the same.

**third exercise**

In a construction it is necessary to install an oil tank whose weight is 400 kg and its volume is 1600 m³.

The machine that will move the tank can only transport objects whose density is less than 1/3 kg/m³. Will the machine be able to carry the oil tank?

When applying the definition of density, the density of the oil tank is:

ρ= 400kg / 1600 m³ = 400/1600 kg/m³ = 1/4 kg/m³.

Since 1/4 < 1/3, it is concluded that the machine can transport the oil tank.

**fourth exercise**

What is the density of a tree whose weight is 1200 kg and its volume is 900 m³?

This exercise only asks to calculate the density of the tree, that is:

ρ= 1200kg / 900m³ = 4/3kg/m³.

Therefore, the density of the tree is 4/3 kilograms per cubic meter.

**Fifth exercise**

I have a pitcher glass and a tube glass. I want to know which of them has higher density.

The jar glass weighs 50 g and has a volume of 200 cm³, while the tube glass weighs 75 g and has a volume of 150 cm³. For this problem ρ1 will be the density of the jar and ρ2 the density of the highball glass.

ρ1= 50/200 g/cm³ = 1/4 g/cm³

ρ2= 75/2000 g/cm³ = 1/2 g/cm³.

Therefore, the highball glass has a higher density than the pitcher glass.

**sixth exercise**

What is the density of an object that has a mass of 300 g in 15 cm³?

Divide the mass by the volume to get the density:

300/15 g/cm³ = 20 g/cm³

So the object has a density of 20 g/cm³

**References**

Barragan, A., Cerpa, G., Rodríguez, M., & Núñez, H. (2006). *Physics for Cinematic High School.* Pearson Education.

Ford, KW (2016). *Basic Physics: Solutions to the Exercises.* World Scientific Publishing Company.

Giancoli, D.C. (2006). *Physics: Principles with Applications.* Pearson Education.

Gomez, AL, & Trejo, HN (2006). *PHYSICS l, A CONSTRUCTIVIST APPROACH.* Pearson Education.

Serway, RA, & Faughn, JS (2001). *Physical.* Pearson Education.

Stroud, KA, & Booth, DJ (2005). *vector analysis* (Illustrated ed.). Industrial Press Inc.

Wilson, JD, & Buffa, AJ (2003). *Physical.* Pearson Education.