How to Calculate How Far a Hollow Cylinder Rolls Up an Incline

How can we determine the distance a hollow cylinder (hoop) will roll up an incline based on its initial speed and the incline's angle?

The distance a hollow cylinder will roll up an incline can be determined using the conservation of energy principle. Kinetic energy is converted to gravitational potential energy, and through calculations involving the object's speed and the incline's angle, the vertical height achieved can be found. Exact computation requires knowledge of the cylinder's mass and radius, which are not provided in the question.

Conservation of Energy Principle

The conservation of energy principle states that the total energy of a closed system remains constant over time. In this scenario, the initial kinetic energy of the hollow cylinder rolling on a horizontal surface is transformed into gravitational potential energy as it moves up an incline.

Calculating Vertical Height

To determine how far up an incline a hollow cylinder will go, we need to apply the principle of conservation of energy. When the cylinder rolls up the incline, its initial kinetic energy is converted into gravitational potential energy. The kinetic energy (KE) of a rolling object is the sum of its translational KE (½mv²) and its rotational KE (½Iω²). For a hollow cylinder, the moment of inertia (I) equals mR², where R is the radius.

Conservation of Energy Equation

The conservation of energy equation for this scenario is:
  • Initial KE = Final potential energy (PE)
  • ½mv² + ½Iω² = mgϒ
  • where g is the acceleration due to gravity, and ϒ is the vertical height achieved on the incline

Solving for Vertical Height

By simplifying and solving the conservation of energy equation using the given values (v = 3.6 m/s, g = 9.8 m/s², and an incline of 17 degrees), we can determine the vertical height the cylinder will reach on the incline. However, an exact numerical answer cannot be calculated without knowing the specific mass and radius of the cylinder.
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