A uniform circular disk, of mass M and diameter D, is initially at rest and subs

Question

A uniform circular disk, of mass M and diameter D, is initially at rest and subsequently experiences a net torque along its z-axis depicted in the graph below. Calculate the disk’s final angular speed, if the axis of rotation is located along the axis perpendicular to the plane of the disk and   

A) through its center of mass. (Express your answer using only the variables M and/or D)   

B) a distance away from its center of mass. (Express your answer using only the variables M, D and or d

C) Calculate the numerical answers to parts (a) and (b) using the following values: M= 2 kg, D= 12 cm, d=2.7 cm

PROBLEM 2: A uniform circular disk, of mass M and diameter D, is initially at rest and subsequently experiences a net torque along its z-axis depicted in the graph below Calculate the disk's final angular speed, if the axis of rotation is located along the axis perpendicular to the plane of the disk and (Nm) 0.2 0.1 0 -0.1 -0.2 2 4 6810 12

Explanation / Answer

radius = R = D/2

axis at the center

moment of inertia = I1 = (1/2)*M*R^2 = (1/2)*M*D^2/4 = M*D^2/8

total impulse = area under the graph = (1/2)*(0.2*8 - 0.1*4) = 0.6

Impule = I*(wf - Wi)

given wi = 0

therefore

I*wf = 0.6

M*D^2/8*wf = 0.6

wf = 4.8/MD^2 <<---answer

---------------

axis at the center

moment of inertia = I2 = (1/2)*M*R^2 + M*d^2 = (1/2)*M*D^2/4 + Md^2/4

total impulse = area under the graph = (1/2)*(0.2*8 - 0.1*4) = 0.6

Impule = I2*(wf - Wi)

given wi = 0

therefore

I*wf = 0.6

(MD^2/8 + Md^2 )*wf = 0.6

wf = 0.6 / (MD^2/8 + Md^2 )        <<---answer

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part(c)

Wf = 4.8/MD^2 = (4.8)/(2*0.12^2) = 166.67 kg m^2 <<---answer

wf = 0.6 / (2*0.12^2/8 + 2*0.027^2 ) = 118.62 kg m^2 <<---answer

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