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匿名 發問於 科學及數學其他 - 科學 · 7 年前


點解第一同第二唔同?wt’s wrong?

2 個解答

  • 天同
    Lv 7
    7 年前

    Angular acceleration a = dw/dt is a differential, you cannot split it up like an algebraic term and combine with delta(w). This is mathematically incorrect.

    The correct derivation is as follows:

    dW = T.(ds)

    where dW is the work done, T is the torque and ds is the angular displacement.

    Because T = I.a

    where I is the moment of inertia

    Thus, dW = I.a.(ds)

    but a = dw/dt (w is the angular velocity)

    dW = I.(dw/dt).ds

    Using the Chain Rule in calculus, dw/dt = (dw/ds).(ds/dt) = w(dw/ds)

    Hence, dW = I.w.(dw/ds).ds

    dW = I.w.dw

    W = integral { I.w.dw} with limits of integration from wi to wf

    i.e. W = I.[integral{w.dw}]

    W = I.(wf^2 - wi^2)/2 = (1/2)I(wf)^2 - (1/2).I.(wi)^2

    2013-10-13 21:14:18 補充:

    Without using integration, you could use the method below:

    In your equation: W = I.(dw/dt).(dw)

    where (dw) denotes "delta w", and (dt) denotes "delta t".

    Hence, W = I.(dw).(dw/dt)

    2013-10-13 21:19:06 補充:

    sorry...a typing mistake in the equation, it should be:

    W = I.(dw/dt).(ds)

    where (dw) denotes "delta w", and (dt) denotes "delta t" and (ds) denotes "delta theta".

    Hence, W = I.(dw).(ds/dt)

    2013-10-13 21:25:06 補充:


    Be aware that (ds/dt) is NOT the instantaneous angular velocity. It is the AVERAGE angular velocity, as it is "delta theta/delta t", not a differential.

    2013-10-13 21:28:18 補充:


    Hence, (ds/dt) = (wi + wf)/2

    where wi and wf are the initial and final angular velocities respectively, assuming that the angular acceleration is constant.

    Because (dw) = (wf - wi)

    Now, W = I.(dw).(ds/dt) = I.(wf - wi).(wi + wf)/2

    W = I.(wf^2 - wi^2)/2 = (I/2)(wf)^2 - (I/2).(wi)^2

    2013-10-13 21:31:40 補充:

    Therefore, what your mistake is that you have mixed up the two "delta w". The fisrt "delta w" is the difference in angular velocities, (wf - wi).

    2013-10-13 21:36:42 補充:

    The second "delta w", which comes from "(delta theta)/(delta t)", is the average angular velocity, which is (wi + wf)/2. Strictly speaking, it should not be denoted by "delta w", which gives a confusion that it is a "difference in angular velocities".

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  • 匿名
    7 年前

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