A string under a tension of 50.0 N is used to whirl a rock...

A string under a tension of $50.0 \mathrm{N}$ is used to whirl a rock in a horizontal circle of radius $2.50 \mathrm{m}$ at a speed of $20.4 \mathrm{m} / \mathrm{s}$ on a frictionless surface as shown in Figure $P 6.39 .$ As the string is pulled in, the speed of the rock increases. When the string is $1.00 \mathrm{m}$ long and the speed of the rock is $51.0 \mathrm{m} / \mathrm{s}$, the string breaks. What is the breaking strength, in newtons, of the string? (FIGURE CAN'T COPY)

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Key Concepts

Centripetal Force

Centripetal force is the net inward force required to keep an object moving in a circular path. It is calculated using the formula F = mv²/r, where m is the mass of the object, v is its speed, and r is the radius of the circle. This concept is fundamental in understanding circular motion dynamics and determining the forces acting on an object moving in a circle.

Conservation of Angular Momentum

Angular momentum is conserved in a system where no external torque is applied, meaning that the product of an object's mass, its velocity, and the radius of its circular path remains constant during any change in radius. This principle explains how the speed of a rotating object increases as its rotation radius decreases, which is critical when analyzing systems where the radius changes, such as a pulled string in circular motion.

Tension in Circular Motion

In systems involving circular motion, the tension in the string (or force in the connector) provides the necessary centripetal force to maintain the motion of the object. Understanding how tension relates to both the mass of the object and its speed at a given radius is crucial, as exceeding the string's tensile strength will cause it to fail. This concept is key when evaluating the conditions under which the system operates safely and when structural limits are reached.

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