inertia

Examples of inertia in the following topics:

• Moment of Inertia

  • 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 basic relationship between the moment of inertia and the angular acceleration is that the larger the moment of inertia, the smaller the angular acceleration.
  • A brief introduction to moment of inertia (rotational inertia) for calculus-based physics students.
  • Identify a property of a mass described by the moment of inertia

• 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.

• The Physical Pendulum

  • The period of a physical pendulum depends upon its moment of inertia about its pivot point and the distance from its center of mass.
  • In this case, the pendulum's period depends on its moment of inertia around the pivot point .
  • where α is the angular acceleration, τ is the torque, and I is the moment of inertia.
  • The moment of inertia of the rigid rod about its center is:
  • A change in shape, size, or mass distribution will change the 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.
  • The ratio depends on the moment of inertia of the object that's rolling.

• 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 .

• The First Law: Inertia

  • Newton’s first law of motion describes inertia.
  • Sometimes this first law of motion is referred to as the law of inertia.
  • Inertia is the property of a body to remain at rest or to remain in motion with constant velocity.
  • Some objects have more inertia than others because the inertia of an object is equivalent to its mass.

• Conservation of Angular Momentum

  • 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.

• 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.

• Conservation of Energy in Rotational Motion

  • Kinetic energy (K.E.) in rotational motion is related to moment of rotational inertia (I) and angular velocity (ω):

• Rotational Collisions

  • As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum.