An engineering team must select one of three interchangeable wheels to serve as the reaction wheel that stabilizes a small research satellite. Each candidate wheel has the same mass m=2.5kg but a different mass distribution about the axle:
• Wheel I – thin hoop of radius 0.20m
• Wheel II – solid uniform disk of radius 0.20m
• Wheel III – solid uniform sphere of radius 0.20m (machined with a central bore so it can spin about a diameter)
During testing, a prototype wheel is spun up to an angular speed of 160rads−1 and is found to store exactly 6.4×102J of rotational kinetic energy. Unfortunately, the identification label indicating which type of wheel was mounted has fallen off.
(a) Calculate the rotational inertia of the wheel that was tested.
(b) Using standard expressions for rotational inertia of common shapes, determine which of the three candidate wheels (I, II, or III) was tested. Show your quantitative reasoning.
(c) Explain qualitatively why the other two wheels would not store the required energy at the same angular speed. Refer to how mass distribution influences rotational inertia.
(d) If the mission requires that the reaction wheel store at least 1.0×103J at 160rads−1, propose which design change (choose one: increase radius, change mass distribution, increase mass) would be most efficient while keeping the wheel’s diameter under 0.25m. Justify your choice in terms of rotational inertia.