#AP Physics 2: Density - Your Ultimate Study Guide 🚀

Hey there, future physicist! Let's dive into density, a fundamental concept that pops up everywhere in AP Physics 2. This guide is designed to be your go-to resource, especially when you're cramming the night before the exam. Let's make sure you're not just ready, but confident! 💪

#Density: The Basics

#What is Density? 🤔

Density ($\rho$) is a measure of how much "stuff" (mass) is packed into a given space (volume). It's the ratio of an object's mass to its volume. Think of it as how "crowded" the molecules are in a substance.

$\rho = \frac{mass}{volume} = \frac{m}{V}$

• Units: The most common unit for density is kilograms per cubic meter ($kg/m^3$), but you'll also see grams per cubic centimeter ($g/cm^3$).
• Intensive Property: Density is an intensive property, meaning it doesn't change with the amount of substance. A tiny drop of water and a huge lake have the same density. 💡

#Density vs. Mass and Volume

• Mass: How much "stuff" an object has. It's an extensive property (changes with the amount of substance).
• Volume: The amount of space an object occupies. Also an extensive property.

#Visualizing Density

Caption: This image illustrates how density relates to the packing of molecules. Denser materials have more molecules packed into the same volume.

#Calculating Density

Let's get hands-on with some examples! Here's how to use the density formula:

#Example Problem #1

Problem: A metal block has a mass of 50 grams and a volume of 10 cubic centimeters. What's its density?

Solution:

$\rho = \frac{m}{V} = \frac{50 , g}{10 , cm^3} = 5 , g/cm^3$

#Example Problem #2

Problem: A wooden cylinder has a mass of 200 grams and a radius of 2 cm. What's its density? (Assume we know the height is h cm)

Solution:

1. Find the volume of the cylinder: $V = \pi r^2 h = \pi (2 , cm)^2 h = 4\pi h , cm^3$
2. Calculate the density: $\rho = \frac{m}{V} = \frac{200 , g}{4\pi h , cm^3} = \frac{50}{\pi h} , g/cm^3$

#Example Problem #3

Problem: A swimming pool has a volume of 50,000 liters and a mass of 400,000 grams. What is the density of the water?

Solution:

$\rho = \frac{m}{V} = \frac{400,000 , g}{50,000 , L} = 8 , g/L$

#Example Problem #4

Problem: Predict the density of ice relative to water at 0 degrees Celsius, and design an investigation to verify your prediction.

Prediction: The density of ice will be less than the density of water at 0°C.

Explanation: Ice forms a lattice structure with more open space between molecules than liquid water. This makes ice less dense.

Investigation:

1. Materials: Container of water, container of ice, balance, graduated cylinder.
2. Measure: Find the mass and volume of both the water and ice.
3. Calculate: Use $\rho = \frac{m}{V}$ to find the densities.
4. Compare: Check if the density of ice is less than the density of water.

#Specific Gravity: Density in Action

#What is Specific Gravity? 🤔

Specific gravity is the ratio of a substance's density to the density of a reference substance (usually water). It's a relative measure, not an absolute one. It tells us if an object will float or sink in a liquid.

Specific Gravity = $\frac{\text{Density of substance}}{\text{Density of reference substance}}$

• Specific Gravity > 1: Object will sink in water.
• Specific Gravity < 1: Object will float in water.

#Specific Gravity and Submersion

Specific gravity also tells us what fraction of an object will be submerged when floating. If an object has a specific gravity of 0.7, 70% of it will be underwater, and 30% will be above water.

#Example: Sponge in Water

Problem: A sponge has a specific gravity of 0.1 in water. What percentage of the sponge is above the water's surface?

Solution:

1. Submerged Portion: Since the specific gravity is 0.1, 10% of the sponge is submerged.
2. Above Water Portion: 100% - 10% = 90% of the sponge is above the water's surface. 🧽

#

• Confusing Mass and Density: Mass is the amount of "stuff," while density is how tightly that "stuff" is packed. Don't mix them up!
• Units: Always pay attention to units. Make sure they are consistent before calculating density.
• Specific Gravity: Remember that specific gravity is a ratio and is relative to a reference substance.

#

• Density is an intensive property.
• Specific gravity is a ratio of densities.
• Ice is less dense than water.
• Objects float if their density is less than the fluid they are in.

#Final Exam Focus 🎯

#High-Priority Topics

• Density Calculations: Practice using the formula $\rho = \frac{m}{V}$.
• Specific Gravity: Understand how it relates to floating and sinking.
• Density and Buoyancy: Connect density to Archimedes' principle and buoyant force.
• Conceptual Questions: Be prepared to explain density concepts and relationships.

#Common Question Types

• Multiple Choice: Calculating density, comparing densities, and identifying factors that affect density.
• Free Response: Designing experiments to measure density, analyzing data, and explaining phenomena related to buoyancy and specific gravity.

#Last-Minute Tips

• Time Management: Don't spend too long on a single question. Move on and come back if you have time.
• Read Carefully: Pay close attention to the wording of the questions. Look for keywords.
• Show Your Work: For free-response questions, show all your steps clearly. Even if you don't get the final answer right, you can get partial credit for your method.
• Stay Calm: Take a deep breath and trust your preparation. You've got this! 💪

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. A 200 g object has a volume of 100 cm³. What is the density of the object? (A) 0.5 g/cm³ (B) 2 g/cm³ (C) 20 g/cm³ (D) 2000 g/cm³

2. If a block of wood floats in water, what can be said about its density? (A) It is greater than the density of water. (B) It is equal to the density of water. (C) It is less than the density of water. (D) It is not related to the density of water.

3. A solid object has a density of 3 g/cm³. If the object is cut in half, what is the density of each piece? (A) 1.5 g/cm³ (B) 3 g/cm³ (C) 6 g/cm³ (D) It cannot be determined.

#Free Response Question

A student is given a sample of an unknown metal and is asked to determine its density. The student uses a balance to measure the mass of the sample and finds it to be 150 grams. The student then places the sample into a graduated cylinder containing 50 mL of water. The water level rises to 75 mL.

(a) Calculate the volume of the metal sample.

(b) Calculate the density of the metal sample.

(c) If the student were to cut the metal sample in half, what would be the density of each piece? Explain your reasoning.

(d) If the metal sample were placed in a liquid with a density of 10 g/mL, would it float or sink? Explain your reasoning.

#FRQ Scoring Breakdown:

(a) (2 points)

• 1 point for correctly determining the volume change (75 mL - 50 mL = 25 mL).
• 1 point for stating the volume of the metal sample (25 mL or 25 cm³).

(b) (2 points)

• 1 point for correctly using the density formula (density = mass / volume).
• 1 point for correctly calculating the density (150 g / 25 cm³ = 6 g/cm³).

(c) (2 points)

• 1 point for stating the density would remain the same.
• 1 point for explaining that density is an intensive property and does not depend on the amount of substance.

(d) (2 points)

• 1 point for stating the metal would sink.
• 1 point for explaining that the metal's density (6 g/mL) is less than the liquid's density (10 g/mL).