How to Calculate the Ratio of Tension in a Vertical Circular Motion
What is the ratio of the tension in the string when the ball is at the top of its path to when it is at the bottom?
A) TbTt=1
B) TbTt=Lv^2 + g/Lv^2 - g
C) TbTt=Lv^2 - g/Lv^2 + g
D) TbTt=1 - v^2gL
E) TbTt=1 + v^2gL
F) TbTt=Lv^2 - gLt^2
G) TbTt=Lv^2 + gLv^2
Answer:
The ratio of the tension Tt in the string when the ball of mass m is at the top of its path to Tb when the ball is at the bottom of its path is Tb/Tt = (Lv^2 + g) / (Lv^2 - g).
The ratio of tension in a vertical circular motion depends on the forces acting on the ball at different points in its path. Let's break down the calculation step by step to understand how the ratio is derived.
1. At the top of the path, the forces acting on the ball are tension Tt and gravity mg. These forces must provide the centripetal force required to keep the ball moving in a circle, which is mv^2/L. Therefore, Tt - mg = mv^2/L, or Tt = mv^2/L + mg.
2. At the bottom of the path, the forces are tension Tb and gravity mg. Here, the tension needs to be greater than the gravitational force to maintain the circular motion, so Tb - mg = mv^2/L, or Tb = mv^2/L + mg.
3. To find the ratio of Tb to Tt, divide the equation for Tb by the equation for Tt: Tb/Tt = (mv^2/L + mg) / (mv^2/L + mg).
4. Simplifying the equation leads to Tb/Tt = (Lv^2 + g) / (Lv^2 - g).
Therefore, the correct answer is B) Tb/Tt = (Lv^2 + g) / (Lv^2 - g).