Physics · 9th Grade · Momentum and Collisions · Weeks 10-18

Center of Mass

Locating the point that represents the average position of the matter in a system.

• Common Core State Standards
• HS-PS2-1
• HS-ETS1-2

About This Topic

The center of mass of a system is the single point that moves as though all external forces on the system act at that point. For a uniform object, the center of mass is at the geometric center; for objects with uneven mass distribution, it shifts toward the heavier regions and can lie outside the physical boundaries of the object entirely. This concept supports HS-PS2-1 and HS-ETS1-2 in the US NGSS framework and appears frequently in engineering design contexts.

The Fosbury Flop, introduced at the 1968 Mexico City Olympics, is one of the most vivid demonstrations of center of mass in sport. By arching backward over the bar, a high jumper can clear the bar while their center of mass passes below it, meaning less energy is required to achieve the same cleared height. The same principle governs vehicle stability: a low center of mass means a larger margin before rollover, which is why pickup trucks and SUVs handle differently than sedans and why SUV rollover rates prompted NHTSA regulatory changes.

Active learning works well here because center of mass is visual and physical. Students who locate the center of mass of irregular objects by suspending them from different points, or who build and test scale models with varying mass distributions, gain spatial intuition that formulas alone do not build. Connecting the concept to engineering decisions, such as why construction cranes are designed the way they are, gives students a meaningful reason to develop precision in their calculations.

Key Questions

1. Why does a high jumper use the "Fosbury Flop" to clear the bar?
2. How does the location of the center of mass affect the stability of a vehicle?
3. How can the center of mass of an object be located outside of its physical body?

Learning Objectives

• Calculate the center of mass for a system of discrete point masses in one and two dimensions.
• Analyze the motion of a system by considering the motion of its center of mass.
• Explain how the distribution of mass affects the location of the center of mass for an extended object.
• Compare the stability of objects with different center of mass locations and heights.
• Design an experiment to locate the center of mass of an irregularly shaped object.

Before You Start

Vectors and Vector Addition

Why: Students need to be able to add vector quantities to calculate the position of the center of mass in multiple dimensions.

Mass and Density

Why: Understanding how mass is distributed is fundamental to locating the average position of that mass.

Newton's Laws of Motion

Why: The concept of center of mass is directly related to how external forces affect the motion of a system.

Key Vocabulary

Center of Mass	The average position of all the mass in a system; the point where the system would balance perfectly.
Momentum	A measure of an object's motion, calculated as mass times velocity; the center of mass of a system moves with constant momentum if no external forces act on it.
Inertia	The resistance of an object to changes in its state of motion; the center of mass represents the point where inertia is concentrated for the entire system.
Torque	A rotational force; unbalanced torques can cause an object to rotate around its center of mass.

Watch Out for These Misconceptions

Common Misconception

The center of mass of an object is always located inside the object.

What to Teach Instead

For objects with holes, concave shapes, or systems of separated objects, the center of mass can lie in empty space. A uniform ring's center of mass is at the geometric center, which is inside the hole. Lab activities where students find the center of mass of ring-shaped or L-shaped cutouts make this concrete before students encounter the weighted-average formula.

Common Misconception

The center of mass is the same as the geometric center of any object.

What to Teach Instead

The center of mass depends on mass distribution, not just shape. An object with most of its mass at one end has its center of mass near that end, not at the midpoint. Students who calculate the center of mass position for weighted metersticks and verify it by balancing build accurate intuition about how mass distribution shifts the center of mass away from the geometric center.

Active Learning Ideas

Hands-On Lab: Locating Center of Mass by Suspension

Small groups receive irregularly shaped cardboard cutouts of US state silhouettes. They suspend each shape from multiple points, draw the plumb-line for each, and find where the lines intersect to identify the center of mass. For at least one concave state shape, they verify the center of mass lies outside the solid region and discuss what physical meaning that has.

30 min·Small Groups

Think-Pair-Share: Fosbury Flop Analysis

Pairs watch a slow-motion video clip of the Fosbury Flop and sketch the athlete's body position at the moment of peak height. They estimate where the center of mass lies and determine whether it clears the bar. The class then discusses how the same cleared height can require less energy than a traditional straddle jump.

25 min·Pairs

Engineering Challenge: Stable Tower Design

Groups build the tallest possible freestanding tower from index cards and tape, given the constraint that the structure must survive a lateral push. They discuss where the center of mass of each design sits, test their towers, and modify designs based on stability failure analysis before a final competition round.

40 min·Small Groups

Structured Calculation: Vehicle Rollover Threshold

Using provided diagrams of a sedan, SUV, and sports car with labeled dimensions and mass distributions, students calculate the center of mass height and track width for each. They then determine the critical tipping angle for each vehicle and rank them by rollover resistance, connecting their calculations to real NHTSA safety ratings.

30 min·Pairs

Real-World Connections

• Structural engineers use center of mass calculations to ensure the stability of buildings and bridges, preventing collapse under various loads and environmental conditions.
• Automotive designers carefully position the center of mass of vehicles, especially sports cars and trucks, to optimize handling, cornering ability, and rollover resistance.
• Professional athletes, like gymnasts and high jumpers, manipulate their body's center of mass during complex movements to achieve greater heights or execute difficult maneuvers.

Assessment Ideas

Quick Check

Provide students with a diagram of three objects: a solid sphere, a hollow sphere, and a sphere with a dense core. Ask them to label the approximate location of the center of mass for each object and briefly justify their reasoning.

Discussion Prompt

Pose the question: 'Why is it easier to carry a long, thin rod by its center than by one of its ends?' Guide students to discuss how the center of mass relates to balance and the forces required to support an object.

Exit Ticket

Present students with a scenario: a construction worker is lifting a heavy, irregularly shaped beam with a crane. Ask them to write two sentences explaining why knowing the beam's center of mass is crucial for safely lifting it.

Frequently Asked Questions

What is the center of mass in physics?

The center of mass is the weighted average position of all the mass in a system. It is the point at which the entire mass of the system can be considered concentrated for analyzing translational motion. External net forces cause the center of mass to accelerate according to Newton's second law, even if the internal parts of the system rotate or change shape around it.

How can the center of mass be outside of an object?

Any object with a non-convex shape, such as a boomerang, a horseshoe, or a hollow ring, can have its center of mass in empty space rather than in solid material. The center of mass is determined by the weighted average positions of all mass elements. If most mass is arranged in a ring, the average position is at the center of the ring where no material exists.

Why does a high jumper use the Fosbury Flop technique?

By arching the body backward over the bar, the jumper keeps each part of the body below the bar at the moment it crosses, even while clearing it. This means the athlete's center of mass travels on a path that passes below the bar. Because the minimum energy needed is determined by how high the center of mass must rise, this technique requires less energy than styles where the center of mass must travel above the bar height.

How does active learning help students grasp center of mass?

Center of mass becomes intuitive through physical experience. When students balance irregular objects, find suspension lines, and design stable structures, they directly observe how mass distribution determines stability, balance, and tipping thresholds. Pure calculation without these experiences leaves students applying formulas without any physical check on whether the result makes sense. Building stability models connects the abstract concept to engineering decisions with real consequences.