Suppose you could use wheels of any type in the design of a soapbox-derby racer (an unpowered, four-wheel vehicle that coasts from rest down a hill). To conform to the rules on the total weight of the vehicle and rider, should you design with large massive wheels or small light wheels? Should you use solid wheels or wheels with most of the mass at the rim? Explain.
Solution 3DQ Step 1: If a rigid body moving through space,its motion can regard as both the combination of translational motion of center of mass and rotational motion about an axis through center of mass. The objects with smaller moment of inertia will roll down the hill faster.because an object rolling down a hill is essentially converting kinetic energy into potential energy. Since kinetic energy K = 1/2 mv 2 + 1/2I w 2 cm cm Where m mass of object v = velocity of center of mass cm I moment of inertia cm w angular speed of object Step 2: Let the mass of car and wheels be M and mass of each wheel m. The initial total energy at top of incline is Mgh. At the bottom of the incline the total energy is sum of the translational kinetic energy of car and rotational energy of the wheels. K = 1/2 mv cm 2+ 4(1/2I cm) 2 Assuming four wheels by conservation of energy. Mgh = 1/2 mv 2+ 4(1/2I w ) 2 cm cm With no slip w = v/R,where R is radius of wheels. Mgh = 1/2 mv cm 2+ 2I cm /R) Multiply above equation through by 2 and divide through by M,then v + 4I(v /MR ) = 2gh = v (I + 4I/MR ) 2 2 Moment of inertia of wheel I = mR Where = 1/2 for disk-like wheels. = 1for hoop-like wheels. 2 2 v (I + 4m/M) = 2gh = v (M + 4m/M) v = 2gh/(M + 4M/M) 2 v = 2gh(M/M + 4m) Hence maximum speed (M/M + 4m)needs to be as large as possible.