Darth Maul decides that doing problems with constant angular acceleration all th

Question

Darth Maul decides that doing problems with constant angular acceleration all the time isn't very fun. So he decides to think of a harder situation... one that begins to make you think deeply about the calculus you know.

He begins to spin an object such that it has an angular acceleration, as a function of time, of ? = 4.6t2 - 3.9t, where ? is in rad/s2 and t in seconds. Assume that the object starts from rest (?o = 0, ?o = 0 at t = 0).

Evaluate ? and ? at t = 8.0 s.

HINT: You will need to think very carefully about this problem, since you need to do "derivatives in reverse." That is, since ? = d?/dt ... you will need to "guess" a function that when you take its derivative, you will get 4.6t2 - 3.9t. Then, just plug in t = 8.0 sec to evaluate at that point.

Then, just do something similar for ?, except you'll need to recognize that ? = d?/dt ... what can you take the derivative of to get this?

Explanation / Answer

alpha = dw/dt

where alpha is ang accel and w is ang velocity

therefore

alpha dt = dw

so to find dw, we can integrate alpha from t=0 to t=8,

integrating the expression for alpha gives us:

4.6/3 t^3 -3.9/2 t^2 evaluated between 0 and 8;

so alpha = 660.26 rad/s

w = d(theta)/dt,

so integrating w dt gives theta when evaluated between the limits of 0 and 8

hence theta = (4.6/3*4) t^4- 1.95/3*t^3 limit from 0 to 8

so

theta = 1237.33 rad

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