Angular Motion & Rotation Dynamics

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Discusses angular motion kinetics and dynamics.
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Angular Motion & Rotation Dynamics

  1. 1. Copyright Sautter 2003
  2. 2. Angular Motion • Angular motion involves rotation or circular motion. Some elements of circular motion have already been discussed and we will review them here. • Circular motion (rotation) can be measured using linear units or angular units. Angular units refer to revolutions, degrees or radians. • The properties of circular motion include displacement, velocity and acceleration. When applied to rotation the values become angular displacement, angular velocity or angular acceleration. Additionally, angular motion can be measured using frequencies and periods or rotation. • The Greek letters theta (), omega () and alpha () are used to represent angular displacement, angular velocity and angular acceleration
  3. 3.   
  4. 4. AVERAGE =  /  t = (2 + 1) / 2  = o t + ½ t2 i = o + t i = ½ (i 2 - o 2) /  s =  r Vlinear =  r alinear =  r f = 1/ T T = 1 / f 1 revolution = 360 degrees = 2  radians  = 2  f  = 2  / T
  5. 5. Moment of Inertia • All states of motion are subject to the laws of inertia, that is tend to remain at the same rate and in the same directional orientation. • In the case of rotational motion, the angular velocity tends to remain unchanged and the plane of rotation persists. • As you will recall, outside forces can change inertial conditions. In rotation, outside torques must be applied to change an objects rotational inertia. • Torque, as you remember, is a force applied perpendicularly to the center of rotation. • τ = F x r
  6. 6. Moment of Inertia • The tendency of a body to resist changes in its linear state of motion is measured by its mass. • The tendency of a body to resist changes in its rotational state of motion is measured by its moment of inertia. • Moment of inertia involves not just the mass of a rotating object but also the distribution of the mass within the object. • τ = F x r, recall that F = ma, therefore: • τ = ma x r, since a =  r, τ = m  r x r =(mr2 ) • I = mr2 and τ = I
  7. 7. Moment of Inertia • τ = I • Note the similarity to F = ma for linear motion. Instead of a applied force (F) and applied torque (τ) is necessary to provide acceleration. • Instead of mass (m), the moment of inertia (I) determines the resulting acceleration. • Instead of linear acceleration (a), angular acceleration () results from the application of a torque. • Although, the moment of inertia in its simplest form is given as mass times radius squared (mr2), in more complex bodies the value of I must be found by calculus methods (integration) or experimental means.
  8. 8. The Laws of Motion for Rotating Bodies (A summary) • First Law – A body which is rotating tends to keep rotating at the same rate and in the same plane unless acted on by an outside torque. • Second Law – Torque = Moment of Inertia x angular acceleration (τ = I) • Third Law – for every torque there must be an equal but opposite torque.
  9. 9. Moment of Inertia • When the moment of inertia is found by experiment a simplifying technique similar to the center of mass concept is used. • Remember, the center of mass of an object is a point where all the mass of the body could be concentrated to give the same inertial properties as the actual mass distribution of the body. • When describing rotational motion, radius of gyration is used in place of the center of mass concept. • The radius of gyration of a body is the radius of a thin ring of a mass equal to the mass of the body which would give the same rotational characteristics as the actual body.
  10. 10. Sphere I = 2/5 mr2 Cylinder I = 1/2 mr2 Thin Ring I = mr2 Thin Rod I = 1/12 mr2 Rotational Axis Rotational Axis Rotational Axis Rotational Axis I for any object of mass m and radius of gyration rg I = mrg 2
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