question_answer

A stone tied to a string is rotated with a uniform speed in a vertical plane. If mass of the stone is m, the length of the string is r and the linear speed of the stone is v, when the stone is at its lowest point, then the tension in the string will be : (g = acceleration due to gravity)

A) \[\frac{m{{v}^{2}}}{r}+mg\]       

B)        \[\frac{m{{v}^{2}}}{r}-mg\]        

C)        \[\frac{mv}{r}\]               

D)         mg

Correct Answer: A

Solution :

                Key Idea: Tension will be more than the centripetal force at the lowest point for circular motion. For a stone of mass m and length of string r at the point B, weight mg acts vertically downwards, while tension \[{{T}_{B}}\] acts vertically upwards. Their resultant provides the necessary centripetal force, that is                                 \[{{T}_{B}}-mg=\frac{m{{v}^{2}}}{r}\]                 \[\Rightarrow \]  \[{{T}_{B}}=\frac{m{{v}^{2}}}{r}+mg\] This is the tension of string at lowest point. Note: Tension of string at highest point A will be less than that at B. \[{{T}_{A}}+mg\frac{mv_{A}^{2}}{r}\] \[\Rightarrow \]  \[{{T}_{A}}=\frac{mv_{A}^{2}}{r}-mg\]