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.

• Movement at Synovial Joints

  • They produce very little rotation or angular movement of the bones.
  • Angular movements are produced by changing the angle between the bones of a joint.
  • Extension past the normal anatomical position is referred to as hyperextension.
  • Retraction occurs as a joint moves back into position after protraction.
  • Depression is the opposite of elevation and involves moving the bone downward, such as after the shoulders are shrugged and the scapulae return to their normal position from an elevated position.

• 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.