Question

a) A square plate has mass 0.600 kg and sides of length 0.150 m. It is free to rotate without friction around an axis through its center and perpendicular to the plane of the plate. How much work must you do on the plate to change its angular speed from 0 to 40.0 rad/s? Express your answer with the appropriate units.

b) How much work must you do on the plate to change its angular speed from 40.0 rad/s to 80.0 rad/s? Express your answer with the appropriate units.

c) In the situation shown in (Figure 1) and (Figure 2), suppose disk A has moment of inertia IA and initial angular speed ωA, while disk B has moment of inertia IA/4 and initial angular speed ωA/2. Initially disks A and B are rotating in the same direction. What is the final common angular speed of the two disks in w_A? What fraction of the initial rotational kinetic energy remains as rotational kinetic energy after the disks have come to their final common angular speed?

Answer 1

a] Moment of inertia = ma^2/6 = 0.6*0.15^2/6 = 0.00225 kg m^2

work done = change in KE = 0.5 I*(w2^2 - w1^2)

= 0.5*0.00225*(40^2-0)

= 1.8 J

b]   work done = change in KE = 0.5 I*(w2^2 - w1^2)

= 0.5*0.00225*(80^2-40^2)

= 5.4 J

c] Final angular speed = [iA*wA + iA/4*wA/2]/(iA+iA/4) = [iA*wA*9/8]/(5*iA/4) = 9*4 wA/[8*5] = 0.9 wA

Final KE/Initial KE = [5/4 iA*(0.9wA)^2]/[iA*wA^2 + iA/4*(wA/2)^2]

= (5/4*0.9^2)/(1+1/16)

= 0.953