7-Rotational Motion

Slideshows:Rotational Kinematics notes, Rotational Dynamics (Torque, Inertia and acceleration), and Rotational Energy and Momentum Notes

Textbook: Chapters 7 & 8 in Mastering Physics (get online code for registration on about page of google classroom)

Objectives:

  • Describe rotational motion
    • use angular displacement, angular velocity and angular acceleration to describe motion about a fixed axis
    • relate rotational motion to linear motion using the radius
    • use + or or - for counterclockwise or clockwise motion.
  • Torque Basics
    • Use representations of the relationship between force and torque.
      • Compare torques on an object caused by forces
      • Estimate torque on an object
    • In equilibrium
      • Design an experiment & analyze data testing a question about balanced torques in a rigid body system
      • Calculate torques in 2D while in static equilibrium (from a diagram or model)
    • Center of mass
      • analyze the motion of a system of 2 objects based on their center of mass
        • in the case of rotation, a force going through the center of mass causes no torque.
  • Unbalanced torques
    • Net torque causes acceleration
      • calculate angular acceleration from the net torque and rotational inertia of an object or rigid system
      • predict angular velocity changes when forces cause a net torque on an object rotating on a fixed axis.
    • Net Torque causes a change in momentum
  • Angular Momentum
    • basics
      • Use right hand rule to find direction of angular momentum (classify as + or -)
      • relate angular momentum to linear momentum by using the radius
      • relate angular momentum to rotational inertia and angular velocity
      • Consider the locations and velocities of individual parts of a system to describe & calculate the angular momentum of a system
        • In particular, calculate momentum for a system of a point mass and extended object
    • changing momentum
      • relate change in angular momentum to average torque and the amount of time the torque is applied.
        • zero net torque indicates momentum is conserved
          • changes in mass distribution can affect the angular speed of a system without changing momentum
          • Consider elliptical orbits in earth-satellite system
      • collisions can cause exchanges of angular momentum
        • predict the effects of a collision in terms of rotational behaviors by finding total momentum before and after a collision.
      • multiple objects rotating together can be treated as a single system with a combined momentum
      • find the change of angular momentum in unfamiliar circumstances & plan labs

Equations:

In addition to the equations above (on the formula chart), you should memorize or be able to recreate the following:

  • use r to convert between angular and linear quantities (v = butt r, a = fish r, etc)
  • Replace angular quantities in linear equations -- like v2 = v02 + 2a(x-x0).
  • Create simple definitions for average angular quantities, like average angular acceleration = change in angular velocity divided by time
  • Know that I = mr2 for point objects at the edge of a circle or hoops, I = 1/2 mr2 for disks or cylinders, and I = 2/5mr2 for spheres.

Common Misconceptions:

    • Misconception: Any force acting on an object will produce a torque. Principle: Force components perpendicular to the radius (or forces with moment arms) cause torque Reasoning: Torque is a measurement of a tendency to change rotation and in order to do that a force must in a way where the line of action of the force does not pass through the center of mass of a free moving object or the pivot point of a fixed object. If it does pass through these points it will affect the whole object equally and not cause rotation, where if it acts to one side or other of the pivot or center, it will cause that side to move along with the force while the other side pivots around the other way.
    • Misconception: Objects moving in a straight line can not have angular momentum. Principle: If an object can give angular momentum to a second object, it must have angular momentum to begin with. Reasoning: Many systems/objects can be evaluated multiple ways. Straight line motion, if considered from a point of rotation that matters for the system, can can be described in terms of how many radians/s it moves as opposed to m/s. While the angular velocity will be increasing as it gets closer to the pivot, the r value will be decreasing and as a result the overall angular momentum of the object a remains constant L=mvr (as shown in the diagram where p=mv and D is the shortest R of the path. This is only meaningful if the angular momentum is going to be transferred to a second object; otherwise it is not intuitive.
    • Misconception: Torque is the same as force and is in same direction.
    • Principle: Torque is a tendency to rotate caused by force with a direction that is 90 degrees to the force and moment arm as determined by the right hand rule.
    • Reasoning: The direction of torque is not terribly important until you talk about changing angular momentum, in which case we see weird stuff like gyroscopic precession (google it and be amazed). This behavior is explained most directly by assigning the direction for angular momentum and torque according to the right hand rule.
    • Misconception: The direction of angular momentum is in direction of linear momentum. Principle: The direction of angular momentum is determined by the right hand rule. Reasoning: Use the same diagram as above for either angular speed or angular momentum, but alter it by using your fingers to represent the movement ov the rotating object, as opposed to the force acting on it. It will always be a direction that is 90 degrees to the plane of rotation, which is weird, but required for things like gyroscopic precession to make sense.

External Resources