If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Rotational inertia and angular second law review

Overview of the key terms, equations, and skills related to rotational inertia, including how to analyze rotation inertia and how it relates to Newton's second law.

Key terms

Term (symbol)Meaning
Rotational inertia (I)Resistance to change in rotational velocity around an axis of rotation. Proportional to the mass and affected by the distribution of mass. Also called the moment of inertia. Scalar quantity with SI units of kgm2.

Equations

EquationSymbolsMeaning in words
α=τnetIα is angular acceleration, τnet is the net torque, and I is the rotational inertiaAngular acceleration is proportional to net torque and inversely proportional to rotational inertia.

Analyzing rotational inertia

Rotational inertia depends both on an object’s mass and how the mass is distributed relative to the axis of rotation. Unlike other scenarios in physics where we simplify situations by pretending we have a point mass, the shape of an object determines its rotational inertia. We can’t just consider the mass to be concentrated at its center of mass.
When a mass moves further from the axis of rotation it becomes more difficult to change the rotational velocity of the system. For example, if we compare the rotational inertia for a hoop and a disc, both with the same mass and radius, the hoop will have a higher rotational inertia because the mass is distributed farther away from the axis of rotation.
Figure 1: A disc and a hoop with the same mass and radius.
If two objects have the same shape but different mass, the heavier one will have a larger moment of inertia.

How does rotational inertia relate to Newton’s second law?

Newton’s 2nd law relates force to acceleration. In the angular version of Newton’s 2nd law, torque τ takes the place of force and rotational inertia takes the place of mass. When the rotational inertia of an object is constant, the angular acceleration is proportional to torque.
Fnet=maτnet=Iα
For example, if we attach a rotating disc to a massless rope and then pull on the rope with constant force, we can see that the angular acceleration of the disc will increase as the force (and the torque) increases. A graph of the angular acceleration vs. torque would have a positive and constant slope because angular acceleration α is directly proportional to torque τ. (See figure 2 below)
Figure 2: Applied torque vs. angular acceleration

Common mistakes and misconceptions

  1. People sometimes forget that angular acceleration can be zero. If the torques on an object cancel out, the net torque is zero and the angular acceleration is also zero. For example, a beam that can rotate about its axis has two forces exerted on it and therefore two torques (see figure 3 below). Since the torques are in opposite directions, the net torque is zero and the beam will not rotate.
Figure 3: A birds-eye view of a horizontal beam parallel to the ground that can rotate about its central axis, with two forces exerted on it.
  1. Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about any axis.

Learn more

For deeper explanations of rotational inertia, see our video on the rotational version of Newton's second law.
To check your understanding and work toward mastering these concepts, check out our exercises:

Want to join the conversation?

  • piceratops ultimate style avatar for user Alberto
    For anyone out there also having trouble understanding why you must get farther to increase the acceleration.

    Alpha = Torque / Moment of Inertia

    Torque = Force * radius (radius of applied force)
    Moment of Inertia = mass * radius^2 (radius of the object).

    However, for a beginner like me it's very easy to think that r in Torque is the same as the r in Moment of Inertia, because of the simple mistake that both are written as r in the formula, and there is no clarification in the notes section.
    (19 votes)
    Default Khan Academy avatar avatar for user
  • spunky sam blue style avatar for user nadavshmila34
    given that the formula for the angular acceleration is:
    alpha = torque/mr^2.
    Under the assumption that the force is applied in a 90 degree angle it can be simplified to:
    F*r/mr^2 which leads to: alpha = F/mr.
    Why is it that in order to get the highest angular acceleration the force must applied at the largest distance from the pivot point when the acceleration is inversely proportional to the distance.

    thanks in advance.
    (6 votes)
    Default Khan Academy avatar avatar for user
  • leafers tree style avatar for user Joci Faubert
    How are the units supposed to work out? How is a (kg*m^2)(rad/s^2) = n/m?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Jahmasen Osaigbovo
    so I took the practice: Angular acceleration and angular second law, and I kept failing. The correction was "Applying the force farther from the axis of rotation increases the angular acceleration, so we should decrease the distance instead." Meanwhile, in the video, David said that an increase in r, results in a decrease in angular acceleration. Why the different school of thoughts?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user gm
    "Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about *any axis.*" What does this mean? If the axis changes, doesn't the torques also changes since r changes in the equation torque=r Fnet?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • mr pink red style avatar for user Perceverance  Nkomo
    It is false that "if the net force on an object with fixed pivot is zero,then its net torque is also zero.

    What is the reason?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • male robot donald style avatar for user Mahir
      The statement is true.
      If the net force is zero, the object is at rest because there is no unbalanced force acting upon it. Torque is force times distance (or radius). If the force is zero, the torque is also zero because they are directly proportional to each other.
      (2 votes)
  • mr pink red style avatar for user V J
    if you have a bar fixed to the wall at a point, will the torque be greater if you push the bar at a point farther away from the wall (fulcrum)?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user anamahamed457
    Can you please explain the following statement,Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about any axis
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user bishaltalukdar66
    How angular acceleration increases if we increase the radius of location the of force? Will it not increase the moment of inertia?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • spunky sam red style avatar for user Arjun Roy
    What exactly is the relationship between an object's rotational acceleration and mass?
    (1 vote)
    Default Khan Academy avatar avatar for user