A solid cylinder rolls without slipping down an incline starting from rest. At the same time a box starts from rest at the same altitude and slides down the same incline with negligible friction. Which arrives at the bottom first?
Question
A solid cylinder rolls without slipping down an incline starting from rest. At the same time a box starts from rest at the same altitude and slides down the same incline with negligible friction. Which arrives at the bottom first?
Solution
The box will reach the bottom first. Here's why:
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The cylinder rolling down the incline has two forms of kinetic energy: translational (due to its movement down the slope) and rotational (due to its spinning). The energy to spin the cylinder comes from the same gravitational potential energy that is also causing the cylinder to translate down the slope.
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The box sliding down the incline only has translational kinetic energy. All of its gravitational potential energy is converted into translational kinetic energy.
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Because the cylinder's gravitational potential energy is split between translational and rotational kinetic energy, it has less energy to translate down the slope compared to the box. Therefore, it moves slower and reaches the bottom after the box.
This assumes that the cylinder is a solid, homogeneous cylinder and that the incline is the same for both the box and the cylinder.
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