moment of inertia

Examples of moment 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

• 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 forces of the two fingers would cancel.

• Applications of Multiple Integrals

  • As is the case with one variable, one can use the multiple integral to find the average of a function over a given set.
  • Given a set $D \subseteq R^n$ and an integrable function $f$ over $D$, the average value of $f$ over its domain is given by:
  • where $m(D)$ is the measure of $D$.
  • In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
  • In the following example, the electric field produced by a distribution of charges given by the volume charge density $\rho (\vec r)$ is obtained by a triple integral of a vector function:

• Rotational Collisions

  • During a collision of objects in a closed system, momentum is always conserved.
  • What if an rotational component of motion is introduced?
  • As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum.
  • For example, take the case of an archer who decides to shoot an arrow of mass m1 at a stationary cylinder of mass m2 and radius r, lying on its side.
  • The arrow hits the edge of the cylinder causing it to roll.

• 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.

• Conservation of Angular Momentum

  • The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.
  • This is an expression for the law of 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.

• Conservation of Energy in Rotational Motion

  • Sparks are flying, and noise and vibration are created as layers of steel are pared from the pole.
  • The force is parallel to the displacement, and so the net work (W) done is the product of the force (F) and the radius (r) of the disc (this is otherwise known as torque(τ)) times the angle (θ) of rotation:
  • Kinetic energy (K.E.) in rotational motion is related to moment of rotational inertia (I) and angular velocity (ω):
  • However, the energy is never destroyed; it merely changes form from rotation of the grindstone to heat when friction is applied.
  • Conclude the interchangeability of force and radius with torque and angle of rotation in determining force

• The First Law: Inertia

  • Newton’s first law of motion describes inertia.
  • The acceleration of an object is parallel and directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.
  • 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.