Angular position

Examples of Angular position in the following topics:

• Angular Position, Theta

  • In mathematics, the angle of rotation (or angular position) is a measurement of the amount (i.e., the angle) that a figure is rotated about a fixed point (often the center of a circle, as shown in ).
  • If $\Delta \theta$ = 2π rad, then the CD has made one complete revolution, and every point on the CD is back at its original position.

• Angular Acceleration, Alpha

  • Angular acceleration is the rate of change of angular velocity, expressed mathematically as $\alpha = \Delta \omega/\Delta t$ .
  • Angular acceleration is the rate of change of angular velocity.
  • Angular acceleration is defined as the rate of change of angular velocity.
  • The units of angular acceleration are (rad/s)/s, or rad/s2.
  • If $\omega$ increases, then $\alpha$ is positive.

• Rotational Angle and Angular Velocity

  • Angular acceleration gives the rate of change of angular velocity.
  • The angle, angular velocity, and angular acceleration are very useful in describing the rotational motion of an object.
  • When the axis of rotation is perpendicular to the position vector, the angular velocity may be calculated by taking the linear velocity $v$ of a point on the edge of the rotating object and dividing by the radius .
  • The object is rotating with an angular velocity equal to $\frac{v}{r}$.
  • The direction of the angular velocity will be along the axis of rotation.

• Angular vs. Linear Quantities

  • It has the same set of vector quantities associated with it, including angular velocity and angular momentum.
  • The units of angular velocity are radians per second.
  • Just as there is an angular version of velocity, so too is there an angular version of acceleration.
  • Just like with linear acceleration, angular acceleration is a change in the angular velocity vector.
  • The blue vector connects the origin (center) of the motion to the position of the particle.

• Simple Harmonic Motion and Uniform Circular Motion

  • This angle is the angle between a straight line drawn from the center of the circle to the objects starting position on the edge and a straight line drawn from the objects ending position on the edge to center of the circle.
  • where the angular rate of rotation is ω.
  • The point P travels around the circle at constant angular velocity ω.
  • To see that the projection undergoes simple harmonic motion, note that its position x is given by:
  • Substituting this expression for ω, we see that the position x is given by:

• 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.
  • The velocity (i.e. angular velocity) is indeed constant.
  • This is the first advantage of describing uniform circular motion in terms of angular velocity.
  • For the length of the arc subtending angle " at the origin and "r" is the radius of the circle containing the position of the particle, we have $s=r\theta $.
  • For the description of the motion, angular quantities are the better choice.

• Moment of Inertia

  • In this way, the moment of inertia plays the same role in rotational dynamics as mass does in linear dynamics: it describes the relationship between angular momentum and angular velocity as well as torque and angular acceleration .
  • A general relationship among the torque, moment of inertia, and angular acceleration is: net τ = Iα, or α = (net τ)/ I.
  • Such torques are either positive or negative and add like ordinary numbers.
  • As can be expected, the larger the torque, the larger the angular acceleration.
  • 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.

• Constant Angular Acceleration

  • Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
  • Simply by using our intuition, we can begin to see the interrelatedness of rotational quantities like θ (angle of rotation), ω (angular velocity) and α (angular acceleration).
  • Similarly, the kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
  • As in linear kinematics where we assumed a is constant, here we assume that angular acceleration α is a constant, and can use the relation: $a=r\alpha $ Where r - radius of curve.Similarly, we have the following relationships between linear and angular values: $v=r\omega \\x=r\theta $
  • Relate angle of rotation, angular velocity, and angular acceleration to their equivalents in linear kinematics

• Rotational Collisions

  • In a closed system, angular momentum is conserved in a similar fashion as linear momentum.
  • For objects with a rotational component, there exists angular momentum.
  • Angular momentum is defined, mathematically, as L=Iω, or L=rxp.
  • An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.
  • After the collision, the arrow sticks to the rolling cylinder and the system has a net angular momentum equal to the original angular momentum of the arrow before the collision.

• 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.
  • The conserved quantity we are investigating is called angular momentum.
  • The symbol for angular momentum is the letter L.
  • If the change in angular momentum ΔL is zero, then the angular momentum is constant; therefore,
  • (I: rotational inertia, $\omega$: angular velocity)