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From 天同: Angular acceleration a = dw/dt is a differenti...
https://hk.answers.yahoo.com/question/index?qid=20131013000051KK00156
https://hk.answers.yahoo.com/question/index?qid=20131013000051KK00156
Sun, 13 Oct 2013 17:08:28 +0000
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
20131013 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)
20131013 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)
20131013 21:25:06 補充：
(cont'd)...
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.
20131013 21:28:18 補充：
(cont'd)...
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
20131013 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).
20131013 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".