If a mass attached to the center of a vertical circle swings around at a fixed speed (v) and gravity pulls straight downward, would the tensions in the rope attached to the mass be different at the top of the circle, straight down to bottom of circle, and directly straight out from the center of the circle? Would they equal? If not, which one has the most and least tension?
For a mass m attached to a string of length L and swings horizontally with a tangential speed of v, the horizontal component of the tension of the string is given by:
T=mv²/r
If it swings in a vertical, this tension has to be adjusted for the weight of the mass due to gravity. At the top, the tension is reduced by mg, (g=acceleration due to gravity) while at the bottom, the weight is additive to the tension.
To determine the tensions in the rope at different positions along the vertical circle, we need to consider the forces acting on the mass. In this case, there are two main forces at play: the gravitational force and the tension in the rope.
At the top of the circle, the tension in the rope will be at its minimum. This is because at this point, the gravitational force is acting downward, trying to pull the mass away from the center of the circle. The tension in the rope provides the necessary centripetal force to keep the mass moving in a circle. So, at the top, the tension in the rope will be equal to the centripetal force required to keep the mass moving in a circle.
As the mass moves from the top of the circle to the bottom, the tension in the rope will gradually increase. This is because the gravitational force is no longer opposing the centripetal force provided by the tension. Instead, it adds to the tension to maintain the circular motion. Therefore, the tension in the rope will be the sum of the centripetal force and the gravitational force acting on the mass.
At the bottom of the circle, the tension in the rope will be at its maximum. This is because, at this point, the gravitational force is acting in the same direction as the centripetal force provided by the tension. The tension in the rope needs to be strong enough to counteract the weight of the mass and maintain circular motion.
Finally, if we consider the position where the mass is directly straight out from the center of the circle (horizontal position), there will still be tension in the rope. In this case, the gravitational force acts perpendicular to the direction of motion, and the tension in the rope provides the necessary centripetal force to maintain the circular path. The tension will be less compared to the bottom of the circle, but greater than at the top.
In summary, the tensions in the rope will be different at different positions along the vertical circle. The tension will be lowest at the top of the circle, gradually increase as the mass moves towards the bottom, reach its maximum at the bottom, and then decrease as the mass moves towards the horizontal position.