The concepts of inertia and momentum are often confused—possibly due to the similarity of their definitions. Inertia is generally described as an object’s resistance to motion, with momentum being the tendency of an object to continue moving. Both have implications for linear motion applications, but while inertia is a fundamental sizing parameter, momentum isn’t directly addressed in system calculations. To distinguish between the two and find out why that is, we’ll look at the definitions and uses of each.

#### Inertia: Resistance to change in speed

Inertia is a body’s resistance to change in speed and is related to its mass and the distance of that mass from the axis of rotation. The classic illustration of inertia is a figure skater spinning on the ice. When her arms are outstretched, a part of her mass is far from the axis of rotation, and therefore she spins at a relatively slow speed. But if she pulls her arms in close to her body, her rate of spin increases, because her entire mass is now close to the axis of rotation.

*I = mr ^{2}*

*I = mass moment of inertia (kg·m ^{2} or lb·ft^{2})*

*m = mass (kg or lb)*

*r = distance from axis of rotation (m or ft)*

*(Note that this is a general equation for the inertia of a point mass. Specific equations are available for various shapes, such as hollow cylinder, solid cylinder, disc, etc.)*

#### Momentum: Mass in motion

Momentum, on the other hand, is the product of an object’s mass and velocity, and is sometimes referred to as “mass in motion.” While a change in shape—the distance of mass from the axis of rotation—will change a system’s inertia, the momentum of a system cannot be changed unless an external force acts on it. This principle is known as the conservation of momentum. The classic example of momentum is a game of billiards. Think of a moving ball, such as the cue ball, colliding with a non-moving ball. If the cue ball stops moving (v=0), its momentum has been completely transferred to the second ball. If the collision results in both balls moving, then the cue ball’s momentum is shared by the two balls.

The equation of momentum for a linear system is simply:

**P = mv**

*P = momentum (kg·m/s or lb·ft/s)*

*m = mass (kg or lb)*

*v = velocity (m/s or ft/s)*

This equation neatly correlates with the earlier description of momentum as “mass in motion.” But when the motion is rotational, the distance of the mass from the rotational axis comes into play. Therefore, angular momentum is expressed as the product of rotational inertia and angular velocity.

*L = I ω*

*L = angular momentum (kg·m ^{2}/s or lb·ft^{2}/s)*

*I = rotational moment of inertia (kg·m ^{2} or lb·ft^{2})*

*ω = angular velocity (rad/s)*

For motion applications, inertia is an important factor in motor sizing calculations. If the motor’s inertia is significantly smaller than the inertia of the load or system, the motor will have difficulty driving and controlling the load, and response time and resonance will be high. Conversely, if the motor inertia is much greater than the load or system inertia, then the motor is likely oversized, and the system will be inefficient.

Although momentum is not directly considered when sizing motion components, its effect is evident. Back to the ice skater example: it’s the conservation of angular momentum principle that dictates that the skater’s speed must increase when her arms are pulled in close to her body. By reducing her inertia (I=mr^{2}, where r has been decreased), her angular velocity, ω, must increase in order for the angular momentum to remain constant.

George Steele says

We were taught simply to use the word inertia when referring to a tendency of a still body to remain at rest – its resistance to acceleration – and momentum when referring to the resistance of a moving body to come to rest or slow down.- its resistance to deceleration. But that is in common English usage, as opposed to strict scientific definition. In principle, however, inertia is the resistance of a body to change in velocity in a frame of reference – whether from 0 velocity to some value along a vector line, or from some value towards 0 in that frame of reference. The fact that there is any confusion in usage is an indictment reflecting the the scientific community’s lack of clarity in meaning.