A body of mass m is tied with string and rotated in vertical circle of radius r. The body is given the velocity u at lowest point What is the tension at the highest point and does the difference in the tensions at the lowest and the highest point depend upon u ? - Wired Faculty

Question

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A body of mass m is tied with string and rotated in vertical circle of radius r. The body is given the velocity u at lowest point $space space left parenthesis u greater than square root of 5 g r right parenthesis end root.$ What is the tension at the highest point and does the difference in the tensions at the lowest and the highest point depend upon u?

Solution

Tension in the string at any point is given by,
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$equals m over r left parenthesis u squared minus 2 g r plus 3 r g cosθ right parenthesis$
At highest point, $straight theta equals 180 degree$

Therefore,

$T subscript H equals m over r left parenthesis u squared minus 2 g r minus 3 r g right parenthesis space equals space m over r left parenthesis u squared minus 5 g r right parenthesis$
At the lowest point, $straight theta equals 0 degree$
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∴ $T subscript L minus T subscript H equals 6 m g$
The difference in the tension at the highest and the lowest point does not depend upon velocity at the lowest point and is constant, equal to 6 mg.

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