The small car, which has a mass of 20 kg, rolls freely on the horizontal track and carries the 5

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The small car, which has a mass of 20 kg, rolls freely on...

The small car, which has a mass of $20 \mathrm{kg},$ rolls freely on the horizontal track and carries the $5-\mathrm{kg}$ sphere mounted on the light rotating rod with $r=$ $0.4 \mathrm{m} .$ A geared motor drive maintains a constant angular speed $\dot{\theta}=4$ rad/s of the rod. If the car has a velocity $v=0.6 \mathrm{m} / \mathrm{s}$ when $\theta=0,$ calculate $v$ when $\theta=60^{\circ}$. Neglect the mass of the wheels and any friction.

Key Concepts

Rotational Kinematics

Rotational kinematics deals with quantities like angular speed and the radius at which a point rotates, determining the tangential velocity of that point. In problems where a maintained constant angular speed is involved, knowing how to convert rotational motion into linear components is key to relating rotational parameters to overall system behavior.

Conservation of Linear Momentum

In a system where no external horizontal forces act, the total linear momentum remains constant. This principle is crucial since, despite internal interactions between parts of the system (in this case, between the car and the rotating sphere), the overall momentum in the horizontal direction does not change over time.

Relative Motion in Rigid Bodies

When analyzing systems with rotating parts mounted on a moving base, it is important to recognize that the total velocity of each component is the vector sum of the translational velocity of the system and the velocity due to rotation about the system’s center of mass. This concept allows us to correctly combine motions occurring in different parts of the system.

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