Equations: Solve the problems, showing all work, including any formulas used. Ma

Question

Equations: Solve the problems, showing all work, including any formulas used. Make it clear for the grader to follow the steps. Partial credit is given for partially correct work or explanation. Circle your final answer. To slow down a spinning centrifuge, it is suddenly immersed in a liquid. If it is initially spinning at an angular velocity of and the angular acceleration it experiences is given by a = where all the constants have positive values. How long does it take to reduce its speed to one half of the initial?

Explanation / Answer

angular acceleration (alpha) = dw/dt

where w is angular speed in rad/s

dw/dt = -Aexp(-Bt)

dw = -Aexp(-Bt) *dt

integrate both sides

w = (A/B) exp(-Bt)

for limits of integral

at t = 0 w = w

at t =t w = w/2

w/2 -w = (A/B) (exp(-Bt) - 1)

w/2 = (A/B) (1-exp (-Bt))

w/2 * (B/A) = (1-exp (-Bt))

exp (-Bt) = (1-( w/2 * (B/A))

-Bt = ln((1-( w/2 * (B/A)))

t = (-1/B ) * ln((1-( (w/2) * (B/A)))

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