Rotational kinetic energy and angular momentum - Rotational Motion

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Rotational kinetic energy and angular momentum



In this lesson, we will learn:

  • How to calculate rotational kinetic energy?
  • Definition of angular momentum
  • Newton’s 2nd law for rotation
  • The law of conservation of angular momentum


Rotational Kinetic Energy

  • Objects rotating about an axis possess “Rotational Kinetic Energy”
  • Objects moving along a straight line possess “Translational Kinetic Energy”.
  • Translational kinetic energy is calculated using, 12mv2\frac{1}{2}mv^{2} , we can convert this formula to rotational kinetic energy question using the rotational motion analogues:

  • Rotational Kinetic Energy and Angular Momentum

    If we consider a rotating ball, every point on the ball is rotating with some speed.
    The ball is made up many tiny particles, each of mass “mm”. let’s take “rr” to be the distance of any one particle from the axis of rotation (O);

    Rotational Kinetic Energy and Angular Momentum

    \Rightarrow \, rotational KE=12Iω2KE = \frac{1}{2}I\, \omega^{2}

  • The center of mass of a rotating object might undergo translational motion (a sphere rolling down an incline), in this case we have to consider both rotational and translational kinetic energy.

  • KE=12Mv2+12ICM  ω2KE = \frac{1}{2}M v^{2} \, + \, \frac{1}{2}I_{CM \;} \omega^{2}

    MM : total mass of the object
    ICMI_{CM} : moment of inertia about the axis through center of mass 
    vv : translational speed 
    ω\omega : angular speed 

    Angular Momentum
    In the previous section, we used the rotational analogue to translate the translational kinetic energy to rotational kinetic energy.
    In like manner, the linear momentum can be changed to angular momentum, using the rotational analogues;

    Rotational Kinetic Energy and Angular Momentum

    LL: is the angular momentum with a standard unit of kg.m2/s

    Newton’s 2nd Law for Rotation

    The Newton’s 2nd law also can be written in terms of rotational analogues;

    F=ma=\sum F = \, ma \, = mΔvΔt\large \frac{m \Delta v}{\Delta t} == ΔpΔt\large \frac{\Delta p}{\Delta t} \quad , in translational motion “Force” causes linear acceleration

    \qquad \quad Similarly for rotational motion , =\sum = ΔLΔt \large \frac{\Delta L }{\Delta t}

    τ=\sum \tau = \, ΔLΔt\large \frac{\Delta L}{\Delta t} = = IΔωΔt\large \frac{I \Delta \omega}{\Delta t} =I\, = I \propto \quad , in rotational motion “Torque” causes rotational acceleration

    F=ma\sum F = ma \quad Newton’s 2nd Law in Translational Motion
    τ=I\sum \tau = I \propto \quad Newton’s 2nd Law in Rotational Motion

    The law of Conservation of Angular Momentum

    The law of conservation of angular moment states that;

    “The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.”

    τ=0    Li=Lf    Iiωi=Ifωf\sum \tau = 0 \; \Rightarrow \; L_{i} = L_{f} \; \Rightarrow \; I_{i \omega i} = I_{f \omega f}

    Example: A skater doing a spin on ice, illustrating conservation of angular momentum.

    Rotational Kinetic Energy and Angular Momentum

    Open arms: Li=Iωi=mri2ωi \qquad L_{i} = I \omega_{i} = mr^{2}_{i} \omega_{i}
    Closed arms:   Lf=Iωf=mrf2ωf\qquad L_{f} = I \omega_{f} = mr^{2}_{f} \omega_{f}

    When the arms of the skater are tucked in, the mass is not changing, but the radius of rotation decreases; rfr_{f} < rir_{i} , therefore;  mrf2mr^{2}_{f} < mri2mr^{2}_{i}

    According to the law of conservation of angular momentum; Li=LfL_{i} = L_{f}

    Li=LfL_{i} = L_{f}

    ωf\qquad \Rightarrow \quad \omega_{f} > ωi  \omega_{i} \; “the result would be higher angular velocity”

    mrf2 mr^{2}_{f} < mri2 mr^{2}_{i}
  • Intro Lesson
  • 2.
    A sphere of radius 24.0 cm and mass of 1.60 kg, starts from the rest and rolls without slipping down a 30.0° incline that is 12.0m long.
  • 4.
    A 2.6kg uniform cylindrical grinding wheel of radius 16cm makes 1600rpm.
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Rotational kinetic energy and angular momentum

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