91Ó°ÊÓ

Angular speed is a measure of how quickly an object moves along a circular path. In this exercise, the object moves with an angular speed of \(6.28\,\text{rad/s}\). Angular speed is different from linear speed, which is how fast an object moves in a straight line.
The formula to calculate angular speed \(\omega\) is:

• \(\omega = \frac{\theta}{t} \)

where \(\theta\) is the angle in radians that the object covers in time \(t\). This means angular speed tells you how many radians the object travels through in a unit of time.
Angular speed helps determine various aspects of circular motion, like the centripetal force needed to maintain the object’s path in a circle. For instance, if you increase the angular speed while keeping the radius constant, you'll need more force to keep the object moving in its circular path.

Tension is the force exerted by a string or rope pulled tight by forces acting from opposite ends. In our exercise, the tension in the string must not exceed \(105\,\text{N}\). This tension is what keeps the object moving in its circular path and prevents it from flying outward.
When we talk about tension, we’re often concerned with the maximum value that the string can handle without breaking. The tension in this scenario is derived from the centripetal force formula:

• \( T = m r \omega^2 \)

Where \(T\) is the tension, \(m\) is the mass of the object, \(r\) is the radius, and \(\omega\) is the angular speed. As the radius decreases, the tension must increase to provide the centripetal force needed for smaller circles.
It is important to calculate this accurately in real-world scenarios, because exceeding the tension limit can result in the string breaking, which could cause the object to fly off its intended path.

Circular motion refers to the movement of an object along the circumference of a circle or rotation along a circular path. For the object in this exercise, circular motion is maintained by a force directed toward the center of the circle, known as the centripetal force.
As the object moves in its path, several forces act upon it:

• The centripetal force pulls the object inward to maintain its path.
• The tension in the string provides the necessary centripetal force.

Without these forces, the object would not continue its circular path and instead move off in a straight line due to inertia. It is crucial in circular motion to keep track of both the forces and the speed to ensure that the object can safely maintain its course.

The radius calculation is key in understanding the limits of circular motion, especially when maximum tension is a constraint, as it is in this example. Given:

• The maximum tension, \(T = 105\,\text{N}\)
• The object’s mass, \(m = 0.500\,\text{kg}\)
• The angular speed, \(\omega = 6.28\,\text{rad/s}\)

You can calculate the smallest radius \(r'\) the object can maintain without breaking the string by rearranging the tension formula:\\[ r' = \frac{T}{m \omega^2} \] Calculating further, substituting in the values:

• \( r' = \frac{105\,\text{N}}{0.500\,\text{kg} \times (6.28\,\text{rad/s})^2} \)
• \( r' = 5.3263\,\text{m} \)

This radius represents the smallest circle the object can move in without the string breaking due to reached tension limits. This understanding aids in safety and design in practical scenarios involving rotations and swings.