Chapter 11

Rotational Mechanics

11.1 Torque
  • Lever Arm: The distance from the turning axis to the point of contact
  • Torque: A force exerted from the axis of rotation of an object
  • Torque = force X lever arm
    • Torque is applied when you want to an object turn or rotate
  • If a force is located on a line that passes through the axis of rotation it will cause the object to move in a straight line instead
  • No force passing through the axis of rotation can cause any torque
  • The same torque can be produced by a large force with a short lever arm or a small force with a long lever arm

11.2 Balanced Torques

  • Rotational Equilibrium: The sum of the torques in a clockwise direction are equal to the sum of the torques in a counter clockwise direction
    • For example, on a seesaw, the distance two people sit from the pivot point is as important as their weight. The heavier person sits a shorter distance from the fulcrum or turning axis,and the lighter person sits farther away. If the torque that produces clockwise rotation by the heavier person equals the torque that produces a counterclockwise rotation by the lighter person, balance will be achieved.
        • Therefore, two people on a non moving seesaw or a balance scale would be in rotational equilibrium

11.3 Torque and Center of Gravity

  • Any force passing through the center of gravity of an object will tend to move the object instead of rotating it
  • If the force instead passes some distance from the center of gravity,then the object will experience a torque that will tend to cause the object to rotate as well as move.

11.4 Rotational Inertia

  • Rotational Inertia: The reluctance of an object to change its state of rotation, determined by the distribution of the mass of the object and the location of the axis of rotation. This is also known as moment of inertia.
  • Just as an object with larger mass is more difficult to start or stop moving than one with a small mass, an object with a large moment of inertia is more difficult to start or stop rotating than one with a small moment of inertia.
  • An object rotating about an axis tends to keep rotating about that axis or rotating objects tend to keep rotating, while non rotating objects tend to stay non rotating.
  • rotational inertia of an object is not always a fixed quantity, it is greater when the mass within the object is extended from the axis of rotation.
    • For example, it is easier to walk or run with your legs bent than straight; bent legs are easier to swing back and forth because it reduces rotational inertia.
  • The farther the mass of an object is from its axis of rotation the larger the object’s moment of inertia, an object with the same mass closer to the axis of rotation will have a smaller moment of inertia
    • Formulas for Rotational Inertia:
      • when all the mass (m) of an object is concentrated at the same distance (r) from a rotational axis then the rotational inertia I=mr²
  • Since objects with large moments of inertia are more resistant to changing their rotational speed than those with smaller moments of inertia, if two or more objects are rolled down an inclined plane the first one to reach the bottom will be the one with the smallest moment of inertia, a solid cylinder will beat a ring and a sphere will beat both the cylinder and the ring.
  • A solid cylinder will roll down a hill faster than a hollow one even if they have the same mass or diameter
  • Objects with the same shape but different size will accelerate equally when rolled down a slanted surface.

11.5 Rotational Inertia and Gymnastics

A human body can be considered to have three principal axes which are at right angles to the others or are mutually perpendicular:
  • longitudinal axis
  • median axis
  • transverse axis
    • each of these axes passes through the center of gravity of the body and the rotational inertia differs about the axis.
    • A body’s moment of inertia is least around the longitudinal axis and about equal around the other two.
Longitudinal Axis:
  • rotational inertia is least around the longitudinal axis (head-to-toe axis) because most of the body's mass is concentrated in this location...this is the easiest rotation to perform
  • rotational inertia is increased by an ice skater going into a spin by extending the leg or an arm. when both arms are extended the rotational inertia is 3 times more than when the arms are tucked in towards to body. By going into a spin with outstretched arms, then draw the arms in, you will have tripled your spin rate.
Transverse Axis:
  • to rotate about your transverse axis you perform a somersault or flip.
  • rotational inertia is at its greatest when the axis is through the hands like when a gymnast does a sommersault on the floor or swings from a horizontal bar with her body fully extended.
  • rotation transfers from one axis to another, from the bar to a line through her CG and she increases her rate of rotation by up to 20 times.
Median Axis:
  • the median axis, or front-to-back axis is used when performing a cartwheel and is a less common axis of rotation

11.6 Angular Momentum

  • a nything that rotates keeps on rotating until it is stopped by another force.
    • linear momentum: product of the mass and the velocity of the object
      • linear momentum = mass X velocity
    • angular momentum: product of rotational inertia and rotational velocity = mvr = Iw
      • angular momentum = rotational inertia X rotational velocity
    • rotational velocity: when a direction is assigned to rotational speed. it is a vector whose magnitude is the rotational speed.
  • Just as an object moving in a straight line has momentum equal to the product of its mass times its velocity, an object that is rotating has angular momentum equal to the product of its moment of inertia times its angular speed.
  • Unlike linear momentum, which is always the same for any mass and velocity, the angular momentum of an object will vary depending on the arrangement of the object’s mass in addition to the mass and angular velocity.

11.7 Conservation of Angular Momentum

  • Law of Conservation of Angular Momentum: If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant.
    • in short, this means that if there is no external torque then the product of rotational inertia and rotational velocity will be the same as at any other time
  • Angular momentum is conserved, a rotating object will always have the same amount unless the angular momentum
    is transferred to some other object
  • This, in combination with the dependence on the arrangement of mass will allow an object to change its rotational speed if the arrangement of mass changes, for example, if you stand on a rotating platform while holding weights starting with your arms outstretched, you will gain speed when you pull your arms in despite your angular momentum remaining constant.
  • conservation of angular momentum formula: mv1r1 = mv2r2