Suppose a solid uniform cylinder rolls without slipping down an inclined plane (like in Figure 12.42). Once the object starts rolling then the rotational kinetic energy of the cylinder

Question

BlackTom AI · Accepted Answer

To analyze the rotational kinetic energy of a solid uniform cylinder rolling without slipping, we use its moment of inertia and the rolling constraint.
Option 1: 'is less than its translational kinetic energy' — For a solid cylinder, I = (1/2) mR^2 and the no-slip condition gives v = ωR. The translational kinetic energy is (1/2) m v^2, while the rotational kinetic energy is (1/2) I ω^2 = (1/2) * (1/2 mR^2) * (v^2/R^2) = (1/4) m v^2. Since (1/4) m v^2 < (1/2) m v^2, the rotational energy is indeed less than the translational energy.
Option 2: 'is equal to its translational kinetic energy' — This would require (1/4) m v^2 = (1/2) m v^2, which is false for a solid cylinder; the rotational part is smaller by a factor of 2.
Option 3: 'is greater than its translational kinetic energy' — As shown above, rotational energy is (1/4) m v^2, which is not greater than (1/2) m v^2; this option contradicts the actual relationship.
Option 4: 'is less than its translational kinetic energy, but only at the start when it is rolling more slowly' — The inequality (rotational < translational) holds for all speeds under the rolling without slipping condition for a solid cylinder; it does not depend on the speed being low or high, so citing a start-only caveat is misleading.
Option 5: 'It is not possible to decide without more information' — We do have enough information: for a solid cylinder with no slipping, the relation is fixed by I and the rolling constraint; no extra data is required to compare KE components.
In summary, the only statement that correctly describes the relationship for a solid uniform cylinder rolling without slipping is that the rotational kinetic energy is less than the translational kinetic energy. The other options either misstate the comparison or add unnecessary caveats.

Suppose a solid uniform cylinder rolls without slipping down an inclined plane (like in Figure 12.42). Once the object starts rolling then the rotational kinetic energy of the cylinder 单项选择题

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