How to Solve: Density Problems

How to Solve: Density Problems

Introduction

"If you can solve density problems, you can figure out why ships float, how to tell real gold from fake, and even pass your physics or chemistry exam—let’s get started!

What You Need To Know First

Before tackling density problems, you must understand: 1. Mass vs. Weight – Mass is the amount of matter (measured in grams or kilograms). Weight is mass × gravity (measured in newtons). 2. Volume – The space an object takes up (measured in cm³, mL, or m³). 3. Units & Conversions – How to convert between grams and kilograms, cm³ and m³, etc.

Key Vocabulary

Formulas To Know

1. Density Formula

Formula: [ \text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)} ]

Variables: - (\rho) (rho) = density (g/cm³ or kg/m³) - (m) = mass (g or kg) - (V) = volume (cm³, mL, or m³)

MEMORISE THIS – It’s the foundation of all density problems.

2. Volume of a Regular Object (Cube, Sphere, Cylinder)

Formulas: - Cube: ( V = \text{side}^3 ) - Rectangular Prism: ( V = \text{length} \times \text{width} \times \text{height} ) - Sphere: ( V = \frac{4}{3} \pi r^3 ) - Cylinder: ( V = \pi r^2 h )

Given on exam sheet? Usually, but memorise the cube and rectangular prism ones.

3. Volume by Displacement (Irregular Objects)

Formula: [ V_{\text{object}} = V_{\text{final}} - V_{\text{initial}} ]

When to use: When an object doesn’t have a regular shape (e.g., a rock).

Step-by-Step Method

Follow these steps for EVERY density problem:

1. Read the question carefully. Underline what’s given and what’s asked.
2. Write down the formula: (\rho = \frac{m}{V}).
3. List the known values (mass, volume, or density).
4. Check units. Convert if needed (e.g., g → kg, cm³ → m³).
5. Rearrange the formula if needed (e.g., ( V = \frac{m}{\rho} ) or ( m = \rho \times V )).
6. Plug in the numbers and solve.
7. Add units to your final answer.
8. Check if the answer makes sense (e.g., density of water is 1 g/cm³—if your answer is 100 g/cm³, something’s wrong).

Worked Example Using the Steps

Question: A metal block has a mass of 200 g and a volume of 25 cm³. What is its density?

Solution: 1. Given: Mass ((m)) = 200 g, Volume ((V)) = 25 cm³ 2. Formula: (\rho = \frac{m}{V}) 3. Plug in numbers: (\rho = \frac{200 \text{ g}}{25 \text{ cm}^3}) 4. Calculate: (\rho = 8 \text{ g/cm}^3) 5. Answer: The density of the metal is 8 g/cm³.

Why this works: We used the basic density formula and made sure units matched.

Worked Examples

Example 1 – Basic (Direct Application)

Question: A liquid has a mass of 150 g and a volume of 120 mL. What is its density?

Solution: 1. Given: (m = 150 \text{ g}), (V = 120 \text{ mL})
(Note: 1 mL = 1 cm³, so units are compatible.) 2. Formula: (\rho = \frac{m}{V}) 3. Plug in: (\rho = \frac{150 \text{ g}}{120 \text{ cm}^3}) 4. Calculate: (\rho = 1.25 \text{ g/cm}^3) 5. Answer: The density is 1.25 g/cm³.

What we did and why: - We used the density formula directly. - We confirmed that mL and cm³ are the same, so no conversion was needed.

Example 2 – Medium (Missing Volume)

Question: A gold bar has a mass of 2 kg and a density of 19.3 g/cm³. What is its volume?

Solution: 1. Given: (m = 2 \text{ kg}), (\rho = 19.3 \text{ g/cm}^3)
(Problem: mass is in kg, density in g/cm³—convert first!) 2. Convert mass: (2 \text{ kg} = 2000 \text{ g}) 3. Rearrange formula: (V = \frac{m}{\rho}) 4. Plug in: (V = \frac{2000 \text{ g}}{19.3 \text{ g/cm}^3}) 5. Calculate: (V \approx 103.63 \text{ cm}^3) 6. Answer: The volume is 103.63 cm³.

What we did and why: - We spotted the unit mismatch (kg vs. g) and converted first. - We rearranged the formula to solve for volume.

Example 3 – Exam Style (Displacement Method)

Question: A student places a rock in a graduated cylinder containing 50 mL of water. The water level rises to 75 mL. The rock’s mass is 100 g. What is its density?

Solution: 1. Find volume by displacement:
(V_{\text{rock}} = V_{\text{final}} - V_{\text{initial}} = 75 \text{ mL} - 50 \text{ mL} = 25 \text{ mL})
(1 mL = 1 cm³, so (V = 25 \text{ cm}^3)) 2. Given: (m = 100 \text{ g}), (V = 25 \text{ cm}^3) 3. Formula: (\rho = \frac{m}{V}) 4. Plug in: (\rho = \frac{100 \text{ g}}{25 \text{ cm}^3}) 5. Calculate: (\rho = 4 \text{ g/cm}^3) 6. Answer: The density of the rock is 4 g/cm³.

What we did and why: - We used displacement to find the volume of an irregular object. - We confirmed that mL and cm³ are equivalent for volume.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—density is just mass divided by volume. Memorise (\rho = \frac{m}{V}), and if you’re missing one variable, rearrange the formula. Always check units—convert kg to g or cm³ to m³ if needed. For irregular objects, use displacement: final volume minus initial volume. Watch out for hidden unit traps and remember that water’s density is 1 g/cm³. If an object’s density is less than that, it floats. Now go crush those density problems!