Λsolid cylinder of radius R is spinned and then placed on an...

$\Lambda$ solid cylinder of radius $R$ is spinned and then placed on an incline having coefficient of friction $\mu=\tan \theta$. 'lhe cylinder continues to spin without falling for time $t=?$ (a) $\frac{R \omega_{n}}{3 g \sin \theta}$ (b) $\frac{R \omega_{o}}{2 g \sin \theta}$ (c) $\frac{R \omega_{o}}{g \sin \theta}$ (d) $\frac{2 R \omega_{o}}{g \sin \theta}$

$\Lambda$ solid cylinder of radius $R$ is spinned and then placed on an incline having coefficient of friction $\mu=\tan \theta$. 'lhe cylinder continues to spin without falling for time $t=?$
(a) $\frac{R \omega_{n}}{3 g \sin \theta}$
(b) $\frac{R \omega_{o}}{2 g \sin \theta}$
(c) $\frac{R \omega_{o}}{g \sin \theta}$
(d) $\frac{2 R \omega_{o}}{g \sin \theta}$

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Key Concepts

Rotational Dynamics

Rotational dynamics is the study of how forces (or torques) affect the rotational motion of objects. It applies principles analogous to Newton's laws in linear dynamics, including the relationship between net torque, moment of inertia, and angular acceleration.

Rolling Without Slipping

This concept refers to the condition in which an object rotates in such a way that the point in contact with the surface has zero relative velocity. It creates a kinematic relationship between the linear velocity (or acceleration) of the object's center of mass and its angular velocity (or acceleration), ensuring coordinated translation and rotation.

Friction on Inclined Planes

Friction on inclined planes involves analyzing how static friction interacts with gravitational forces on a slope. The coefficient of friction in such cases often relates directly to the angle of the incline, determining whether an object can remain in static equilibrium, roll without slipping, or begin to slide.

Moment of Inertia

The moment of inertia quantifies a body’s resistance to changes in its rotational motion and depends on the object's mass distribution. In problems involving cylinders and other shapes, using the appropriate moment of inertia is key to correctly applying the equations of rotational dynamics.

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