Classical Mechanics

Rotational Kinetic Energy

Conservation of rotational and translational energy

         

A solid ball rolls on a slope from rest starting from a height of H=7.0 mH=7.0\text{ m} and then rolls on a horizontal region, as shown in the above figure. The horizontal distance of the slope and the distance of the horizontal region are both equal to L=6 m,L=6 \text{ m}, and the height of the horizontal region is h=3.0 m.h=3.0\text{ m}. Approximately how far horizontally from point PP does the ball hit the floor?

The rotational inertia of a solid sphere about any diameter is I=25MR2,I=\frac{2}{5}MR^2, where MM and RR are the mass and the radius of the solid sphere, respectively, and the gravitational acceleration is g=9.8 m/s2.g=9.8\text{ m/s}^2.

A brand new off-road 6×66 \times 6 pickup truck has six wheels. If the total mass of the pickup truck is 1200 kg1200\text{ kg} and the mass of each of the six wheels is 19 kg,19\text{ kg}, what fraction of its total kinetic energy is due to the rotation of the wheels about their center axles?

Assume that each of the six wheels is a uniform disk.

A very thin hoop with a mass of 180 kg180\text{ kg} is rolling along a horizontal floor. If the speed of the hoop's center of mass is 0.180 m/s,0.180\text{ m/s}, how much work must be done on the hoop to stop it?

A solid cylinder with radius 40 cm40\text{ cm} and mass 12 kg12\text{ kg} is rolling down from rest without slipping for a distance of L=10.0 m,L=10.0\text{ m}, as shown in the above figure. The angle of the slope is θ=30.\theta=30^\circ. What is the approximate angular speed of the cylinder about its center when it just meets the bottom of the slope, assuming that the gravitational acceleration is g=9.8 m/s2?g=9.8\text{ m/s}^2?

Consider a situation where a uniform solid sphere, which was rolling smoothly along a horizontal floor, rises up along a 30.030.0^\circ slope. If it stops after it has rolled 9.00 m9.00\text{ m} along the slope and then begins to roll backward, what was its initial speed?

The rotational inertia of a solid sphere about any diameter is I=25MR2,I=\frac{2}{5}MR^2, where MM and RR are the mass and the radius of the solid sphere, respectively, and the gravitational acceleration is g=9.8 m/s2.g=9.8\text{ m/s}^2.

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