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When a mass or object moves in a circle or a curved path, it experiences what is referred to as centripetal acceleration. This type of acceleration is always directed towards the center of the circle and is essential for keeping the object in circular motion. Imagine swinging a ball on a string in a horizontal circle overhead; the string pulls the ball toward the center, preventing it from flying off in a straight line, which is what it would naturally do due to inertia.

In physics, centripetal acceleration (\( a_c \) can be calculated using the formula \( a_c = \frac{v^2}{r} \) or \( a_c = \omega^2 \cdot r \) where \(v\) represents the linear velocity, \( \omega \) is the angular velocity, and \( r \) is the radius of the circular path. Therefore, even with a constant angular velocity, the presence of a curved path necessitates a non-zero centripetal acceleration.

Consider a figure skater performing a spin on ice. As the skater pulls her arms in, she spins faster; this is an example of rigid body rotation. A rigid body is a solid object that doesn’t deform or change shape as it moves—like the tether and masses in the exercise. In such cases, all parts of the body rotate around a fixed axis, and every point in the body maintains a constant distance from that axis.

During rigid body rotation, different points on the object have different linear speeds, but share the same angular velocity. This is because while the angle covered by all points in a given time is the same (due to the constant angular velocity), the distances from the axis are different, leading to varying paths and thus different linear speeds.

If an object rotates at a constant angular velocity, it means its rate of rotation does not change over time. Angular velocity (\( \omega \) is measured in radians per second (rad/s) and provides a way to express the speed of rotation. For instance, a ferris wheel that takes a minute to complete a full revolution has a slower angular velocity compared to a spinning top that completes multiple revolutions in a second.

With a constant angular velocity, although the rate of rotation is steady, it is crucial to understand that it doesn't imply there's no acceleration. If there's a change in the direction of the velocity vector of the rotating object, as seen in the rotating tether with masses, centripetal acceleration will be present, and therefore, total acceleration cannot be zero despite the constant speed of rotation.