Introduction

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This article delves into a fascinating question in physics: do heavier balls roll faster downhill than lighter ones of the same size? By examining Newtonian mechanics, rotational dynamics, and the concept of moment of inertia, we will explore the answer to this intriguing query.

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Understanding the Concept of Moment of Inertia

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The moment of inertia (I) is a critical parameter in determining the motion of rotating objects. It characterizes the mass distribution of an object relative to an axis of rotation. For a solid sphere, the moment of inertia is given by:

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I (frac{2}{5}mr^2)

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Here, m represents the mass of the sphere, and r represents its radius. This equation underscores how the moment of inertia is influenced by both the mass and the radius of the sphere.

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Rolling Without Slipping

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When a sphere rolls without slipping, the linear speed ((v)) at the bottom of the incline is directly related to the angular speed ((omega)) by the formula:

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v r(omega)

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This relationship is fundamental to understanding the dynamics of rolling objects. It highlights how the linear motion of the sphere is linked to its rotational motion.

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Energy Considerations

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At the top of an incline, the gravitational potential energy (mgh) is converted into both translational and rotational kinetic energy at the bottom. The equations that govern this transformation are:

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Translational kinetic energy: (frac{1}{2}mv^2)

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Rotational kinetic energy: (frac{1}{2}Iomega^2)

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Substituting (omega frac{v}{r}) into the equation for rotational kinetic energy:

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(frac{1}{2}Iomega^2 frac{1}{2}(frac{2}{5}mr^2)(frac{v}{r})^2 frac{1}{5}mv^2)

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Combining these, the total mechanical energy at the top is equated to the sum of translational and rotational kinetic energy at the bottom:

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mgh (frac{1}{2}mv^2 frac{1}{5}mv^2 frac{7}{10}mv^2)

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Solving for (v):

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(v sqrt{frac{10gh}{7}})

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Comparison of Larger vs. Smaller Balls

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Consider two balls of the same mass but different radii. The larger sphere has a greater moment of inertia, but its radius is larger as well. The mass of the balls is the same, so the ratio of their speeds is determined by their moments of inertia and the energy distribution.

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Both balls start with the same potential energy at the top and convert it to kinetic energy at the bottom. Since the formula for the final linear speed is the same, regardless of the moment of inertia, both balls will reach the bottom with the same speed:

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(v sqrt{frac{10gh}{7}})

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This conclusion is robust and holds true for identical mass balls rolling without slipping, irrespective of their sizes.

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Conclusion

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In summary, both a larger and a smaller ball of the same mass will roll down a hill at the same speed when rolling without slipping. The differences in their sizes and moments of inertia do not affect the final linear speed, ensuring they reach the bottom simultaneously if released from the same height.

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Key takeaways include the importance of moment of inertia in determining rotational dynamics and the energy transformations involved in the motion of rolling objects.