FOCUSED REVIEW
Focused Review: Center of Mass and Systems of Particles
Reinforce the highest-leverage center-of-mass calculations and system-motion relationships.
TIME
Approximately 15 minutes
BEST FOR
Targeted reinforcement
FINISH WITH
A readiness check
After this focused review, you'll be able to... calculate center of mass for discrete particles, use P = M VCM, and determine center-of-mass acceleration from net external force.
Choose how you want to review
Unit Review Overview
This Physics Sensei Unit Review reinforces the key ideas and problem-solving skills for Center of Mass and Systems of Particles. Use it for homework support, quiz preparation, exam review, or independent study.
UNIT: MEC-U19
TOPIC: Center of Mass and Systems of Particles
TREATMENT: Algebra-Based
LEVEL: Introductory college physics
BEST USED
✓ To reinforce key concepts
✓ Before starting homework
✓ Before a quiz or exam
Physics Sensei is an independent educational resource organized around core college-physics ideas, problem-solving models, and study workflows.
Your Review Plan
Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.
6 Stages • Approximately 15 minutes.
Warm-Up Check
Before you begin, take a moment to see what you already remember. Do not worry about getting everything right. This is only a starting point.
ACTIVITY 1
Center-of-Mass Calculation
Recall the mass-weighted-average formula for center of mass.
For masses 2.0 kg at x = 1.0 m and 3.0 kg at x = 5.0 m, find xCM.
Reveal Answers
xCM = 3.4 m.
Why it works: xCM = [2.0(1.0)+3.0(5.0)]/5.0 = 3.4 m.
ACTIVITY 2
Momentum Relation
Recall the system-level momentum relation.
A system has total mass 5.0 kg and total momentum +10 kg·m/s. Find VCM.
Reveal Answers
VCM = +2.0 m/s.
Why it works: Use P = M VCM, so VCM = P/M = 10/5.0 = +2.0 m/s.
ACTIVITY 3
External-Force Relation
Connect net external force to center-of-mass acceleration.
A 4.0 kg system experiences a net external force of 12 N. Find ACM.
Reveal Answers
ACM = 3.0 m/s2.
Why it works: ΣFext = M ACM, so ACM = 12/4.0 = 3.0 m/s2.
Ready to strengthen your understanding?
You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?
Core Concepts
Reinforce the two highest-leverage relationships, then use them in representative situations.
KEY CONCEPT 1
Discrete Center of Mass
For discrete particles, calculate the center of mass as a mass-weighted average. In two dimensions, evaluate xCM and yCM separately using the same total mass.
xCM = Σ(mixi)/M; yCM = Σ(miyi)/M.
Example: For 2.0 kg at x = 1.0 m and 3.0 kg at x = 5.0 m, xCM = 3.4 m.
Sensei Note: Do not average positions unless the masses are equal. Center of mass is a mass-weighted average.
KEY CONCEPT 2
System Momentum and External Force
Total momentum is connected to center-of-mass velocity by P = M VCM. The acceleration of the center of mass is determined by the net external force: ΣFext = M ACM.
P = M VCM; ΣFext = M ACM.
Example: If M = 5.0 kg and P = +10 kg·m/s, VCM = +2.0 m/s. If a 4.0 kg system experiences 12 N net external force, ACM = 3.0 m/s2.
Sensei Note: Internal forces can redistribute momentum among particles, but only the net external force determines center-of-mass acceleration.
KEY CONCEPT 3
How the Focused Ideas Connect
Calculate the mass-weighted position first. Then use P = M VCM when the question concerns system momentum or velocity, and ΣFext = M ACM when it concerns how the system accelerates.
Focused strategy: weighted position → total momentum → net external force.
Ready to apply these ideas?
You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?
Guided Practice
Apply the reinforced ideas to two representative situations, then use the strategy card to check your setup.
PRACTICE 1
Two-Dimensional Center of Mass
Calculate xCM and yCM using mass-weighted coordinates.
Particles of 1.0 kg, 2.0 kg, and 3.0 kg are at (0,0), (3,0), and (0,4) m. Find xCM and yCM.
Reveal Answers
xCM = 1.0 m; yCM = 2.0 m.
Why it works: The total mass is 6.0 kg. xCM = [1(0)+2(3)+3(0)]/6 = 1.0 m and yCM = [1(0)+2(0)+3(4)]/6 = 2.0 m.
PRACTICE 2
Momentum Check
Find total momentum, then use it to determine VCM.
A 3.0 kg cart moves at +2.0 m/s and a 2.0 kg cart moves at −1.0 m/s. Find P and VCM.
Reveal Answers
P = +4.0 kg·m/s; VCM = +0.80 m/s.
Why it works: P = 3.0(+2.0)+2.0(−1.0)=+4.0 kg·m/s. With M = 5.0 kg, VCM = P/M = +0.80 m/s.
PRACTICE 3
Focused Setup Strategy
Before calculating, identify whether the target is position, momentum/velocity, or acceleration.
Which relation should you choose for each target: center-of-mass position, center-of-mass velocity from total momentum, or center-of-mass acceleration from net external force?
Reveal Answers
Position: xCM or yCM weighted average. Velocity: P = M VCM. Acceleration: ΣFext = M ACM.
Why it works: Choosing the target quantity first prevents mixing particle-level equations with system-level equations.
Ready to check your understanding?
You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?
Confidence Check
You've rebuilt the key ideas and practiced them with guidance. Now try these short questions on your own to check your understanding before moving on.
QUICK CHECK 1
Center-of-Mass Calculation
Use the mass-weighted average, not the geometric midpoint.
Two masses, 4.0 kg at x = 0 and 6.0 kg at x = 5.0 m, form a system. Find xCM.
Reveal Answers
xCM = 3.0 m.
Why it works: xCM = [4.0(0)+6.0(5.0)]/10.0 = 3.0 m.
QUICK CHECK 2
System Motion
Connect total momentum to center-of-mass velocity.
A 10 kg system has VCM = +0.60 m/s. Find the total momentum. If ΣFext = 0, what happens to VCM?
Reveal Answers
P = +6.0 kg·m/s, and VCM remains +0.60 m/s if ΣFext = 0.
Why it works: P = M VCM = 10(0.60)=+6.0 kg·m/s. With zero net external force, total momentum and therefore VCM remain constant.
QUICK CHECK 3
Interpret Your Focused Check
Use the two results above to decide whether to continue or revisit one relationship.
Did you correctly choose between the center-of-mass position, momentum, and external-force relations?
Reveal Answers
If yes, continue. If not, revisit the relation tied to the quantity you missed.
Why it works: Focused review is about recognizing the correct system-level model quickly, not adding more problem volume.
How did it go?
You've checked your understanding. Take one final look at the essential ideas before deciding what to do next. Need a quick reminder?
Summary
Before moving on, take one final look at the most important ideas from this review.
KEY TAKEAWAY 1
Use Mass-Weighted Coordinates
Compute each center-of-mass coordinate from the weighted position sum divided by total mass.
KEY TAKEAWAY 2
Use System-Level Momentum and Force Relations
Use P = M VCM and ΣFext = M ACM to connect particle behavior to system motion.
KEY TAKEAWAY 3
Choose the Relation from the Target Quantity
Position uses a mass-weighted average; velocity uses total momentum; acceleration uses net external force.
Ready for your next step?
You've reviewed the essential ideas one last time. Now choose the resource that best matches how confident you feel. Need a quick reminder?
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