Moment — Definition, Formula & Examples

Moment

A number indicating the degree to which a figure tends to balance on a given line (axis). A moment of zero indicates perfect balance, and a large moment indicates a strong tendency to tip over.

Formally, the moment of a point P about a fixed axis is the mass of P times the distance from P to the axis. For a figure, the moment is the cumulative sum of the moments of all the figure's points. This cumulative sum is the same as the mass of the figure times the distance from the figure's center of mass to the fixed axis.

Note: This is similar to, but not the same as, the physics quantity known as moment of inertia.

1. Plane region bounded above by $y = f(x)\text{,}$ below by the x-axis, on the left by $x = a\text{,}$ and on the right by $x = b\text{.}$ The density of the region is uniform (constant) and equals $\rho\text{.}$ $M_y = \rho\int_a^b x f(x)\,dx \qquad M_x = \rho\int_a^b \frac{1}{2}\big[f(x)\big]^2\,dx$ Note: $M_y$ is the moment about the y-axis. $M_x$ is the moment about the x-axis.

2. Plane region bounded above by $y = f(x)\text{,}$ below by $y = g(x)\text{,}$ on the left by $x = a\text{,}$ and on the right by $x = b\text{.}$ The density of the region is uniform (constant) and equals $\rho\text{.}$ $M_y = \rho\int_a^b x\big[f(x) - g(x)\big]\,dx \qquad M_x = \rho\int_a^b \frac{1}{2}\Big(\big[f(x)\big]^2 - \big[g(x)\big]^2\Big)\,dx$ Note: $M_y$ is the moment about the y-axis. $M_x$ is the moment about the x-axis.

Key Formula

M = \sum_{i=1}^{n} m_i \, d_i

Where:

$M$ = Total moment of the system about the chosen axis
$m_i$ = Mass (or weight) of the i-th point
$d_i$ = Signed distance from the i-th point to the axis
$n$ = Number of points in the system

Worked Example

Problem: Three point masses lie along the x-axis: mass 2 kg at x = 1, mass 5 kg at x = 3, and mass 3 kg at x = 6. Find the moment of this system about the origin (x = 0).

Step 1: Write the moment formula. Each point contributes its mass times its distance from the axis (the origin, x = 0).

M = \sum_{i=1}^{3} m_i \, d_i

Step 2: Compute each individual moment.

m_1 d_1 = 2 \times 1 = 2, \quad m_2 d_2 = 5 \times 3 = 15, \quad m_3 d_3 = 3 \times 6 = 18

Step 3: Sum all individual moments to get the total moment about the origin.

M = 2 + 15 + 18 = 35

Step 4: Verify using the center of mass. The total mass is 10 kg, and the center of mass is at x = 35/10 = 3.5. The moment equals total mass times center of mass distance: 10 × 3.5 = 35. ✓

\bar{x} = \frac{35}{10} = 3.5, \quad M = 10 \times 3.5 = 35

Answer: The moment of the system about the origin is 35 kg·units.

Another Example

This example differs by using signed (negative) distances, showing how moments on opposite sides of the axis partially cancel and how the sign of the total moment indicates which direction the system tips.

Problem: Two point masses lie along the x-axis: mass 4 kg at x = −2 and mass 6 kg at x = 3. Find the moment about the origin and determine which side the system tips toward.

Step 1: Note that distances can be negative (to the left of the axis). The signed distance matters because moment measures the tendency to rotate in a particular direction.

d_1 = -2, \quad d_2 = 3

Step 2: Compute each individual moment using the signed distances.

m_1 d_1 = 4 \times (-2) = -8, \quad m_2 d_2 = 6 \times 3 = 18

Step 3: Sum to get the total moment.

M = -8 + 18 = 10

Step 4: Interpret the result. A positive moment means the system tips toward the positive x-direction. The center of mass is at x = 10/10 = 1, which is to the right of the origin.

\bar{x} = \frac{10}{10} = 1

Answer: The moment about the origin is 10 kg·units. The system tips to the right (positive side).

Frequently Asked Questions

What is the difference between moment and moment of inertia?

A moment (first moment) equals mass times distance from an axis:

m \times d

. Moment of inertia (second moment) equals mass times the square of the distance:

m \times d^2

. The first moment tells you about balance and center of mass, while the moment of inertia tells you how hard it is to make an object rotate. They are related concepts but use different powers of the distance.

When is the moment equal to zero?

The moment about an axis equals zero when the system is perfectly balanced on that axis. This happens when the center of mass lies exactly on the axis. For example, if equal masses are placed at equal distances on opposite sides of the axis, the positive and negative moments cancel to zero.

How is moment related to center of mass?

The center of mass is found by dividing the total moment by the total mass:

\bar{x} = M / m_{\text{total}}

. Put another way, the total moment of a system equals the total mass multiplied by the distance from the center of mass to the axis. So computing moments is the key step in finding the center of mass.

Moment (First Moment) vs. Moment of Inertia (Second Moment)

	Moment (First Moment)	Moment of Inertia (Second Moment)
Definition	Sum of mass × distance from axis	Sum of mass × (distance from axis)²
Formula	$M = \sum m_i d_i$	$I = \sum m_i d_i^2$
Power of distance	First power (linear)	Second power (squared)
What it measures	Tendency to tip; used to find center of mass	Resistance to rotational acceleration
Can be zero or negative?	Yes — zero means balanced, negative indicates direction	Always non-negative (distances are squared)

Why It Matters

Moments appear throughout calculus when you compute the center of mass of a region, lamina, or solid of revolution — these are standard topics in Calculus II. They also form the foundation for torque and equilibrium problems in physics. Understanding moments helps you solve real-world problems like finding the balance point of a beam, designing stable structures, or analyzing weight distributions.

Common Mistakes

Mistake: Forgetting to use signed distances when points lie on opposite sides of the axis.

Correction: Distances to the left or below the axis should be negative. If you take absolute values instead, you will overestimate the moment and get the wrong center of mass.

Mistake: Confusing moment with moment of inertia and squaring the distances.

Correction: The (first) moment uses distance to the first power:

m \times d

. Only use

m \times d^2

when computing moment of inertia. Check which quantity the problem is asking for.

Related Terms

Center of Mass Formula — Found by dividing moment by total mass
Geometric Figure — Object whose moment is computed
Line — The axis about which moment is measured
Point — Basic unit contributing to moment
Sum — Moment is a cumulative sum of contributions
Zero — A moment of zero means perfect balance