The speed of the ball when it is at an angular position of θwith respect to vertical (A) √((10 g

step 1 Dear students, in the given multiple choice question, question 71 of chapter 9, oscillation and

The speed of the ball when it is at an angular position of...

The speed of the ball when it is at an angular position of $\theta$ with respect to vertical (A) $\sqrt{\frac{10 g}{7}[h-r+R)+(R-r) \cos \theta}$ (B) $\sqrt{\frac{10 g}{7}[(h-R)+(R-r) \cos \theta}$ (C) $\sqrt{\frac{5 g}{7}[h-(r+R)+(R-r \cos \theta)}$ (D) $\sqrt{\frac{5 g}{7}[h-(R+r)(1-\cos \theta)}$

The speed of the ball when it is at an angular position of $\theta$ with respect to vertical
(A) $\sqrt{\frac{10 g}{7}[h-r+R)+(R-r) \cos \theta}$
(B) $\sqrt{\frac{10 g}{7}[(h-R)+(R-r) \cos \theta}$
(C) $\sqrt{\frac{5 g}{7}[h-(r+R)+(R-r \cos \theta)}$
(D) $\sqrt{\frac{5 g}{7}[h-(R+r)(1-\cos \theta)}$

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Key Concepts

Conservation of Energy

This fundamental principle states that the total energy in a closed system remains constant. For a rolling ball, gravitational potential energy is converted into kinetic energy (both translational and rotational) as it moves, allowing us to relate changes in height or position to its speed.

Rolling Motion

Rolling motion involves both translation and rotation. When a sphere rolls without slipping, its total kinetic energy is the sum of translational kinetic energy (due to the center of mass moving) and rotational kinetic energy (due to spinning about the center of mass). This combination leads to characteristic energy factors in the equations.

Angular Geometry in Curved Paths

Describing positions using an angular displacement relative to the vertical is common in problems involving curved paths. The cosine of the angle appears in the expression for the vertical displacement, which directly affects the potential energy change and thus the speed of the object.

Moment of Inertia

The moment of inertia measures a body's resistance to rotational acceleration. For a solid sphere, the moment of inertia is a key quantity when calculating the rotational kinetic energy, and it influences how the energy is partitioned between translation and rotation in a rolling motion.

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