A hemisphere of weight W and radius r is released from rest...

A hemisphere of weight $W$ and radius $r$ is released from rest in the position shown. Determine ( $a$ ) the minimum value of $\mu_{s}$ for which the hemisphere starts to roll without sliding, $(b)$ the corresponding the acceleration of point $B$. [Hint: Note that $O G=\frac{3}{8} r$ and that, by the parallel-axis theorem, $\left.\bar{I}=\frac{2}{5} m r^{2}-m(O G)^{2} .\right]$

A hemisphere of weight $W$ and radius $r$ is released from rest in the
position shown. Determine ( $a$ ) the minimum value of $\mu_{s}$ for which
the hemisphere starts to roll without sliding, $(b)$ the corresponding the
acceleration of point $B$. [Hint: Note that $O G=\frac{3}{8} r$ and that, by the
parallel-axis theorem, $\left.\bar{I}=\frac{2}{5} m r^{2}-m(O G)^{2} .\right]$

Key Concepts

Newton’s Second Law for Rotation

Newton’s second law for rotation establishes that the net torque acting on an object is equal to the product of its moment of inertia and its angular acceleration. This principle is used to connect the forces (such as friction) with rotational effects and to derive the motion equations for objects undergoing simultaneous translational and rotational motion.

Parallel-Axis Theorem

The parallel-axis theorem is used to compute the moment of inertia of a body about any axis parallel to one through its center of mass. It allows modifications of the standard moment of inertia by accounting for an additional term that takes into consideration the shifted axis. This theorem is fundamental when the rotation occurs around an axis that is not centered on the mass distribution.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. In rolling problems, it not only prevents slipping but also provides the necessary torque to initiate and sustain rolling motion. The minimum value of the static friction coefficient is critical because it dictates whether the object will roll without sliding under the applied forces.

Rolling Without Slipping

This concept involves ensuring that the point of contact between a rolling object and a surface remains stationary with respect to the surface during motion. It provides the kinematic constraint that links the translational velocity of the center of mass with the angular velocity of the object, which is essential in determining the condition for pure rolling motion without sliding.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the mass distribution relative to that axis. In dynamics problems, especially with complex shapes like a hemisphere, calculating the moment of inertia is essential for determining angular accelerations and understanding the rotational behavior under applied torques.

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