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Angular velocity (\(\boldsymbol{u}\)) describes how fast an object rotates or revolves relative to another point. In other words, it tells us how quickly the angle of the object changes as it moves along a circular path. Mathematically, it is expressed in radians per second (rad/s).
To calculate angular velocity, use the formula: \(\boldsymbol{u = 2\boldsymbol{u} \times \text{{revolutions per second}} \text{{(in radians per second)}}}\).

In the given example, the mass completes 3.5 revolutions per second. Plugging in the values, we get \(\boldsymbol{u = 2\boldsymbol{u} \times 3.5 = 7\boldsymbol{u}} \text{{rad/s}}\). This tells us that the mass has an angular velocity of 7Ï€ radians per second as it moves along its circular path.

Circular motion occurs when an object moves along the circumference of a circle. It can be uniform or non-uniform, depending on whether the object's speed remains constant or changes.

In this problem, the mass undergoes uniform circular motion, completing 3.5 revolutions per second at a constant rate. The key parameter of uniform circular motion is that the object's speed and the angular velocity are constant, which simplifies the calculations.

The radius of the circular path (2.45 meters in this case) is crucial in many calculations involving circular motion, such as finding the centripetal acceleration. 
If you know the angular velocity and the radius, you can easily compute the centripetal acceleration!

Centripetal force is the force that keeps an object moving in a circular path, acting towards the center of the circle. Without it, the object would move in a straight line due to its inertia.

The formula to calculate the centripetal force (\boldsymbol{F_c}) is:
\[F_c = m \times a_c\]
where \boldsymbol{a_c} is the centripetal acceleration and \bold::m:: is the mass of the object. Centripetal force arises from various types of forces:

• Tension in a string
• Gravitational force
• Electromagnetic force

In the solution provided, we've calculated the centripetal acceleration using the angular velocity (\boldsymbol{u}) and the radius (\boldsymbol{r}):\boldsymbol{a_c = u^2 \times r}By substituting \boldsymbol{u = 7u} rad/s and \bold::r:: = 2.45 m, we've derived the centripetal acceleration. Here, it helps to maintain the mass on its circular path.