A car of mass, m, rolls from rest down an incline of height, h. If each cylindrical wheel has an additional mass of (¼)m and radius, r, develop an expression for the car's translational velocity when it reaches the base of the incline.

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Consider the initial and final energy of the system.

The only initial energy is potential energy = Mgh where M=m+4(¼)m=2m for a car, mass m, with 4 wheels, each of mass ¼m.

At the end of the incline the system has gained energy=½mv²+4(½)(I⍵²), that is, kinetic energy in the body of the car and the rotational energy in its wheels. I is the moment of inertia of each cylindrical wheel and ⍵ is the angular velocity of each of the wheels. The car and its wheels move as one down the incline so ⍵=v/r.

Energy must be conserved assuming no heat is generated, so:

2mgh=½mv²+2Iv²/r². But what is I? For solid cylinders I=½mass × radius², and for hollow cylinders=mass × radius². If the wheels are solid cylinders, 2mgh=½mv²+2(½)(¼)mr²(v²/r²)=½mv²+¼mv²=¾mv². The m’s cancel out and 2gh=¾v², so v=√(8gh/3). For hollow cylinders, 2mgh=mv² and v=√(2gh). So the velocity is greater when the wheels are solid.
 

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