The 4 -kg ball and the attached light rod rotate in the...

The 4 -kg ball and the attached light rod rotate in the vertical plane about the fixed axis at $O .$ If the assembly is released from rest at $\theta=0$ and moves under the action of the 60 -N force, which is maintained normal to the rod, determine the velocity $v$ of the ball as $\theta$ approaches $90^{\circ} .$ Treat the ball as a particle.

Key Concepts

Work-Energy Principle in Rotational Motion

This concept states that the net work done on a rotating body is equal to its change in kinetic energy. In problems involving forces acting on objects moving in arcs, such as a ball on a rod, the work done by the applied force (integrated along the path of motion) converts directly into rotational kinetic energy, thus allowing one to solve for the speed if the work and kinetic energy parameters are known.

Particle Motion on a Circular Path

When an object is constrained to move in a circle, its motion can often be analyzed as that of a particle moving along a circular arc. In such cases, the tangential acceleration and speed are related through the radius of the circle. This concept is important for problems where the force always acts perpendicular to the radial line, ensuring that the applied force contributes directly to the tangential (linear) acceleration.

Tangential Force and Work Integration

A key aspect in rotational dynamics is that when a force is applied tangentially, the work done is obtained by integrating the force along the arc of motion. Since the arc length is related to the angular displacement (with s = r?), a constant force applied perpendicularly allows for a straightforward computation of the work done, hence linking the force to the increase in kinetic energy.

Torque and Its Relation to Force

Torque is the rotational equivalent of force and is given by the product of the force and the lever arm (radius). When the force is maintained normal to the rod, the torque is maximized and remains constant. This steady torque causes angular acceleration, and through energy considerations, one can relate this torque to the final speed of the particle moving in the circular path.

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