The angular acceleration of a wheel, as a function of time, is ? =5.0 t 2?8.5 t

Question

The angular acceleration of a wheel, as a function of time, is ?=5.0t2?8.5t, where ? is in rad/s2 and t in seconds.

Part A.

If the wheel starts from rest (? = 0, ?= 0, at t = 0), determine a formula for the angular velocity ?, as a function of time.

Express your answer in terms of t.

Part B.

Determine a formula for the angular position ?, as a function of time.

Express your answer in terms of t.

Part C.

Evaluate ? at t = 3.5s .

Express your answer to two significant figures and include the appropriate units.

Part D.

Evaluate ? at t = 3.5s .

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

a) ? = d? / dt
d? = ? dt

Integrate to find ? as a function of time
?(t) = integral (5.0*t^2 - 8.5*t dt)
?(t) = 5.0/3*t^3 - 8.5/2*t^2 + C1

Initial condition ?(0) = 0
0 = 0 - 0 + C1
C1 = 0

So the equation is just
?(t) = 1.67*t^3 - 4.25*t^2

b) ? = d?/dt
d? = ? dt
Integrate to get theta
?(t) = integral (1.67*t^3 - 4.25*t^2 dt)
?(t) = 1.67 / 4*t^4 - 4.25/3*t^3 + C2

Initial condition ?(0) = 0
0 = 0 - 0 + C2
C2 = 0

?(t) = 0.417*t^4 - 1.417*t^3

c)?(3.5) = 0.417*t^4 - 1.417*t^3

= 0.417*(3.5)^4 - 1.417*(3.5)^3

=3.5^3(0.417*3.5-1.417)

=1.82

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