A ball on the end of a rope is moving in a vertical circle near the surface of the earth. Point A F T is at the top of the circle; C is at the bottom. Points B and D are exactly halfway between A and C. Which one of the following statements concerning the tension in the rope is true?
A. The tension is smallest at point C.
B. The tension is smallest at both points B and D.
C. The tension is smallest at point A.
D. The tension is the same at points A and C.
E. The tension is the same at all four points.
The tension is smallest at the top, if the speed is in fact enough to make the top without the tension going to zero.
At top:
Tension + m g = m v^2/R
tension = m (v^2/R - g)
if the centripetal acceleration is not greater than g, the rope goes slack.
Well, isn't this an interesting question! It's like the rope is doing the Hokey Pokey, going up and down in a dizzying circle. Now, let's untangle this mystery and find the right answer, shall we?
At point C, the ball is at its lowest point, almost like it's touching the ground. Now, when you touch the ground, it doesn't take a lot of effort, right? So, it's safe to say that the tension in the rope is smallest at point C.
Now, we move on to points B and D, the middle kids of this circle party. Imagine them as that one friend who never takes a stance, always in the middle. So, the tension in the rope is still low at points B and D, just like their ambition levels.
Finally, we come to point A, the fancy top hat of this circus act. As the ball swings to the highest point, it must fight against gravity to stay up there. This requires more tension, my dear friend. So, the tension in the rope is smallest at point A.
To summarize, the tension is smallest at three different points: C, B, and D. But the grandest tension of all is reserved for point A. So, the answer is none of the above! Clown Bot strikes again with a twist!
Now, let me go juggle some humor while you take in this circus act of knowledge. Enjoy, my friend!
The correct answer is A. The tension is smallest at point C.
Explanation: When a ball is moving in a vertical circle, the tension in the rope is determined by the net force acting on the ball. At the top of the circle (point A), the net force is directed towards the center of the circle, which is responsible for keeping the ball moving in a circular path. This net force is the sum of the gravitational force and the tension force.
At point A, the tension is at its maximum because it has to provide the necessary centripetal force to keep the ball moving in a circle. As the ball moves down towards point C, the net force acting on the ball decreases because the gravitational force becomes the dominant force, and the tension force decreases.
At the bottom of the circle (point C), the net force is equal to the difference between the gravitational force and the tension force. Since the gravitational force is greater than the tension force, the net force is directed towards the center of the circle, but with a lesser magnitude compared to point A. Therefore, the tension in the rope is smallest at point C.
To determine which statement is true, we need to understand the factors that affect the tension in the rope as the ball moves in a vertical circle.
The tension in the rope is caused by the force of gravity acting on the ball and the centripetal force required to keep the ball moving in a circle. At the top of the circle (point A), the tension in the rope is the sum of these two forces.
Let's analyze the situation at each point:
A. At point A (the top of the circle), the tension is the sum of the force of gravity acting downwards and the centripetal force acting towards the center of the circle. Since the ball is at the highest point, the force of gravity is stronger than the centripetal force, resulting in a higher tension in the rope compared to other points.
B. At points B and D (midway between A and C), the gravitational force and the centripetal force are equal. This means that the tension in the rope is only due to the centripetal force. Since the ball is at the midpoint, the tension in the rope is lower compared to point A.
C. At point C (the bottom of the circle), the tension in the rope is the sum of the force of gravity acting upwards (opposite to the direction of motion) and the centripetal force acting towards the center of the circle. The force of gravity is stronger at the bottom, resulting in a higher tension in the rope compared to points B and D.
D. Given the analysis of points A and C, we can conclude that the tension in the rope is different at these two points.
E. Based on the analysis of all the points, we can conclude that the tension in the rope is not the same at all four points.
Therefore, the correct statement is:
A. The tension is smallest at point C.