Physics Part 3

  1. How are Angular speed and Kinetic Energy related to each other?
    • An object rotating about a fixed axis remains stationnary in space, so there is no kinetic energy associated with translational motion. The rotating particles making up the rotating object, however, are moving through space; they follow circular paths.Image Upload 1
    • If the mass of the ith particle is mi and its tangential speed is vi, its kinetic energy is:
    • Image Upload 2
    • Image Upload 3
    • Image Upload 4
    • Where we define the quantity in the parentheses as the moment of inertia I of a the rigid object.Image Upload 5Units (kg m2)
    • And at last we can define Kinetic Rotational Energy:
    • Image Upload 6
  2. State an example where an object has different rotating parts but dome of them have Rotational Kinetic Energy and some do not.
    • Observe the following picture:
    • Image Upload 7
    • The masses m do not have any Kinetic anergy because they do not travel through space and the masses M rotate around the y=axis
    • There fore we can calculate the Moment of inertia I of the system and Rotational Kinetic Energy in the following way:
    • Image Upload 8
  3. State the calculated Moment of Inertia I (kg m2) for a thin sylindrical shell. (thin hoop)
    Image Upload 9
  4. State the calculated Moment of Inertia I (kg m2) for a Hollow Cylinder.
    Image Upload 10
  5. State the calculated Moment of Inertia I (kg m2) for a Solid cylinder or disk.
    Image Upload 11
  6. State the calculated Moment of Inertia I (kg m2) for a Long, thin rod with rotation axis through center.
    Image Upload 12
  7. State the calculated Moment of Inertia I (kg m2) for a long, thin rod with rotation axis through the end.
    Image Upload 13
  8. State the calculated Moment of Inertia I (kg m2) for a solid sphere
    Image Upload 14
  9. State the calculated Moment of Inertia I (kg m2) for a Thin spherical shell
    Image Upload 15
  10. What is Torque (Image Upload 16)
    When a force is exerted on a rigid object pivoted about an axis, the object tends to rotate about that axis. The tenency of a force to rotate an object about some axis is measured by a quantity called torque (Image Upload 17) Greek letter tau.

    • We define the magnitude of the torque associated with the force around the axis with the expression:
    • Image Upload 18
    • Where r is the distance between the rotation axis and the point of application of the force and d is the perpendicular distance from the rotation axis to the line of action of F.
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    • Total torque on a system is the summation of all the torques.
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  11. Stablish a relationship for Torque and Moment of inertia.
    Image Upload 21

    • Image Upload 22
    • That is, the net tirque acting on the particle is proportional to its angular acceleration, and the proportionality constant
    • is the moment of inertia. Notice that:
    • Summation of Torque = I Image Upload 23 has the same mathematical form as Newton's second law of motion,
    • Summation of Forces = ma

    • There fore wue can stablish the following formula:
    • Image Upload 24Units (Nm)
    • Notice that I depends on the shape of the object. Recall for example a I = 1/2 MR2 for a solid disk.
  12. Rolling with out slipping, explain.
    Rolling with out slipping refers to the fact that the rolling object is never "dragged, instead, the rolling object is incostant rolling motion with the surfice. (never translating)

    • Image Upload 25
    • As we can see in the picture the cylinder rotates through an angle theta, its center of mas CM moves a linear distance s = RImage Upload 26 therefore the speed of the center of mass for pure rolling motion is given by:
    • Image Upload 27
    • And we can also calculate translational acceleration with the following idea:
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  13. State the mathematical formula to calculate the Kinetic Energy of an dylinder rolling with out slipping.
    • Let us start with a picture:
    • Image Upload 29
    • Notice that when the cylinder is rolling with out splipping the cylinder has two different types of kinetic (rolling) energies. One is translational and the other one rotational.

    • In the formula for kinetic energy of a rolling cylinder the two kinetic energies must be consider.
    • Image Upload 30
    • The first term represents the rotational kinetic energy of the objects about its center of mass, and the second term represents the kinetic energy the object would have if it were just translating throug space without totating. Therefore the total kinetic energy is the sum of the two.
  14. Define angular momentum.
    • An important consideration in defining angular momentum is the process of multiplying two vectors by means of the opeation called the vector product. The torque vector Image Upload 31 is related to the two vectors r and F. We can stablish that relationship using a mathematical operation called the vector product. (cross product)
    • Image Upload 32Units (Nm)
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  15. Vector product operation (cross product) is a commutative operation. By this we mean:

    AxB = BxA


    • False
    • False, the cross product operation is NOT commutative.
  16. Define Angular Momentum L.
    • Angular Momentum L is the cross product operation of the position vector r and the linear momentum vector p. Angular momentum is always concerve in a system where is no net external torque.
    • Image Upload 34Units (kg m2/s)
    • Image Upload 35Image Upload 36
    • Where I is the moment of inertia if the object
    • The macguntude of angular momentum is:
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    • We can also add the following formula to keep in mind that the summation of torque in a system is equal to the change in angular momentum over change in time.
    • Image Upload 40
  17. How can we calculate the Net external torque on a system?
    • The net torque on a system equals the time rate of change of angular momentum of the system.
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  18. Identify the feature link to the angular momentum of a rotating rigid object.
    • We can begin by expressing the magnitude of angular momentum as:
    • Image Upload 42
    • Image Upload 43

    • Notice that acording to the picture the disk is rotating aroung the z-axis. Therefore we can state the folowing mathematical formula to find Angular momentum:
    • Image Upload 44
  19. Calculate total Torque on a system using the Moment of inertia of an object and angular acceleration.
    • Rotational form of Newton's Second Law.
    • Image Upload 45
    • That is, the net external torque acting on a rigid object about a fixed axis equals the moment of inertia about the rotation axis multiplied gy the object's angular acceleration relative to that axis.
  20. Angular momentum is always conserved if net external torque acting on the system is zero?

    a) True

    b) False
    a) True

    • The total angular momentum of a system is constant in both magnitude and direction if the external torque acting on the system is zero, that is, if the system is isolated.
    • Image Upload 46
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    • Recall the formula: Image Upload 48
    • Well, in order for angular momentum of a rigid ritating object to be conserved we must have I and Image Upload 49 constant.

  21. List the three quantities that remain constant on certain conditions. These three quanties one could say that are the only energies that of most likely conserved in an isolated system.
    Image Upload 50(if there are no energy tranfers acrros the system boundary.)

    Image Upload 51(if the nt external force on the system is zero)

    Image Upload 52(if the net external torque on the system is zero)
Card Set
Physics Part 3
Physics Part Three