Four uniform objects having the same mass and diameter are released simultaneous

Question

Four uniform objects having the same mass and diameter are released simultaneously from rest at the same distance above the bottom of a hill and roll down without slipping. The objects are a solid sphere, a solid cylinder, a hollow cylinder, and a thin-walled hollow cylinder.


Question 2: Which of the following statements are correct? (There may be more than one correct choice.)
a- All these objects have the same forward speed at the bottom of the hill.
b- All these objects have the same kinetic energy at the bottom of the hill.
c- All these objects have the same rotational kinetic energy at the bottom of the hill.
d- All these objects have the same angular velocity at the bottom of the hill.

Explanation / Answer

a and b are definitely correct, after all, all the objects have vo = 0 and a = -9.8m/s^2. Given this and they're going down the same hill, d is constant for vf^2 = 19.8d (m/s)^2 for all 4 objects. Thus the kinetctic energy is 19.8Md J for all objects, where M is the mass of each object. Were also told that all objects have the same diameter (let's say d). Thus the rotational KE =M(theta)^2 = M(vf/r)^2= 4M(vf/d)^2. Since M, vf, and d are the same for all objects, statement c is true. In this decuction we got theta = 2vf/d, so d is true too. Conclusion, all 4 are correct.

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