A 0.30-kg rock is swung in a circular path and in a vertical plane on a 0.25-m-length string.At the top of the path,the angular speed is 12.0 rad/s.What is the tension in the string at that point?
A)7.9 N
B)16 N
C)18 N
D)83 N

Question

Quizplus · Accepted Answer

The tension in the string at the top of the path can be calculated using the formula for centripetal force, which is $F_c = m \cdot r \cdot \omega^2$, where $m$ is the mass of the rock, $r$ is the radius of the circular path (length of the string), and $\omega$ is the angular speed. However, at the top of the path, the tension in the string must also support the weight of the rock, so the total tension is the sum of the centripetal force and the weight of the rock ($mg$), where $g$ is the acceleration due to gravity (approximately $9.8 m/s^2$).The centripetal force is $F_c = 0.30 \cdot 0.25 \cdot 12^2 = 10.8 N$.The weight of the rock is $F_g = m \cdot g = 0.30 \cdot 9.8 = 2.94 N$.The total tension at the top of the path is the sum of these two forces: $T = F_c + F_g = 10.8 N + 2.94 N = 13.74 N$.However, none of the options exactly match this calculation, suggesting a mistake in my explanation or calculation. Given the options, the closest correct approach to calculate tension at the top involves considering both the centripetal force needed to keep the rock moving in a circle and the gravitational force acting on the rock. The correct formula to find the tension at the top combines these forces correctly, but it seems there was an error in my calculation or in interpreting the options provided. Revisiting the calculation with the correct approach but ensuring accuracy in the application of formulas and consideration of the forces involved would be necessary to match the provided options. Given the discrepancy, let's correct the approach:At the top of the circular path, the tension in the string is actually less because it has to provide the centripetal force required to keep the rock moving in a circle, minus the weight of the rock, since both forces are acting in the same direction (towards the center of the circle). The correct formula to calculate the tension at the top is $T = m \cdot r \cdot \omega^2 - m \cdot g$.Using the given values: $T = 0.30 \cdot 0.25 \cdot 12^2 - 0.30 \cdot 9.8 = 10.8 - 2.94 = 7.86 N$, which rounds to 7.9 N, matching option A. This correction addresses the mistake in the initial explanation and aligns with the principles of circular motion and gravity.

A 030-Kg Rock Is Swung in a Circular Path and in in a Vertical