rotational inertia

(noun)

The tendency of a rotating object to remain rotating unless a torque is applied to it.

Related Terms

Examples of rotational inertia in the following topics:

  • Rotational Inertia

    • Rotational inertia is the tendency of a rotating object to remain rotating unless a torque is applied to it.
    • Rotational inertia, as illustrated in , is the resistance of objects to changes in their rotation.
    • This equation is the rotational analog of Newton's second law (F=ma), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr2 is analogous to mass (or inertia).
    • The quantity mr2 is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of rotation.
    • Different shapes of objects have different rotational inertia which depend on the distribution of their mass.

  • Relationship Between Torque and Angular Acceleration

    • Torque is equal to the moment of inertia times the angular acceleration.
    • Torque and angular acceleration are related by the following formula where is the objects moment of inertia and $\alpha$ is the angular acceleration .
    • If you replace torque with force and rotational inertia with mass and angular acceleration with linear acceleration, you get Newton's Second Law back out.
    • The net torque about an axis of rotation is equal to the product of the rotational inertia about that axis and the angular acceleration, as shown in Figure 1 .
    • With rotating objects, we can say that unless an outside torque is applied, a rotating object will stay rotating and an object at rest will not begin rotating.

  • Relationship Between Linear and Rotational Quantitues

    • The description of motion could be sometimes easier with angular quantities such as angular velocity, rotational inertia, torque, etc.
    • Second advantage is that angular velocity conveys the physical sense of the rotation of the particle as against linear velocity, which indicates translational motion.
    • Alternatively, angular description emphasizes the distinction between two types of motion (translational and rotational).
    • With the relationship of the linear and angular speed/acceleration, we can derive the following four rotational kinematic equations for constant $a$ and $\alpha$:
    • As we use mass, linear momentum, translational kinetic energy, and Newton's 2nd law to describe linear motion, we can describe a general rotational motion using corresponding scalar/vector/tensor quantities:

  • Moment of Inertia

    • The moment of inertia is a property of a mass that measures its resistance to rotational acceleration about one or more axes.
    • Newton's first law, which describes the inertia of a body in linear motion, can be extended to the inertia of a body rotating about an axis using the moment of inertia.
    • Moment of inertia also depends on the axis about which you rotate an object.
    • The moment of inertia in the case of rotation about a different axis other than the center of mass is given by the parallel axis theorem.
    • A brief introduction to moment of inertia (rotational inertia) for calculus-based physics students.

  • Conservation of Angular Momentum

    • Let us consider some examples of momentum: the Earth continues to spin at the same rate it has for billions of years; a high-diver who is "rotating" when jumping off the board does not need to make any physical effort to continue rotating, and indeed would be unable to stop rotating before hitting the water.
    • We can see this by considering Newton's 2nd law for rotational motion:
    • When she does this, the rotational inertia decreases and the rotation rate increases in order to keep the angular momentum $L = I \omega$ constant.
    • (I: rotational inertia, $\omega$: angular velocity)
    • In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia.

  • Rotational Kinetic Energy: Work, Energy, and Power

    • Looking at rotational energy separately around an object's axis of rotation yields the following dependence on the object's moment of inertia:
    • where $\omega$ is the angular velocity and $I$ is the moment of inertia around the axis of rotation.
    • In the rotating system, the moment of inertia takes the role of the mass and the angular velocity takes the role of the linear velocity.
    • The Earth has a moment of inertia, I = 8.04×1037 kg·m2.
    • Express the rotational kinetic energy as a function of the angular velocity and the moment of inertia, and relate it to the total kinetic energy

  • Conservation of Energy in Rotational Motion

    • Energy is conserved in rotational motion just as in translational motion.
    • The simplest rotational situation is one in which the net force is exerted perpendicular to the radius of a disc and remains perpendicular as the disc starts to rotate.
    • Kinetic energy (K.E.) in rotational motion is related to moment of rotational inertia (I) and angular velocity (ω):
    • The final rotational kinetic energy equals the work done by the torque:
    • This confirms that the work done went into rotational kinetic energy.

  • The Coriolois Force

    • The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame.
    • It is proportional to the object's speed in the rotating frame.
    • They allow the application of Newton's laws to a rotating system.
    • The Coriolis effect is caused by the rotation of the Earth and the inertia of the mass experiencing the effect.
    • This effect is responsible for the rotation of large cyclones.

  • Rotational Collisions

    • What if an rotational component of motion is introduced?
    • For objects with a rotational component, there exists angular momentum.
    • As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum.
    • Once the arrow is released, it has a linear momentum p=mv1i and an angular component relative to the cylinders rotating axis, L=rp=rm1v1i.
    • Evaluate the difference in equation variables in rotational versus angular momentum

  • Balance and Determining Equilibrium

    • With hair cells in the inner ear that sense linear and rotational motion, the vestibular system determines equilibrium and balance states.
    • The otolith contains calcium carbonate crystals, making it denser and giving it greater inertia than the macula.
    • The ampulla contains the hair cells that respond to rotational movement, such as turning your head from side to side when saying "no."
    • The difference in inertia between the hair cell stereocilia and the otolith in which they are embedded leads to a force that causes the stereocilia to bend in the direction of that linear acceleration.
    • Rotational movement of the head is encoded by the hair cells in the base of the semicircular canals.