Question

A sphere of mass 'M' is attached to one end of a metal wire having length 'L' and diameter 'D'. It is whirled in a vertical circle of radius R with angular velocity 'ω'. When the sphere is at lowest point of its path, the elongation of the wire is ______.

(Y= Young's modulus of the material of the wire, g =acceleration due to gravity)

Options

• `(4"ML"("R"omega^2+g))/(pi"D"^2"Y")`

• `("ML"("R"omega^2+g))/(2pi"D"^2"Y")`

• `(6"ML"("R"^2omega^2+g))/(pi"D"^2"Y")`

• `(2"ML"("R"^2omega^2+g))/(pi"D"^2"Y")`

Answer 1

A sphere of mass 'M' is attached to one end of a metal wire having length 'L' and diameter 'D'. It is whirled in a vertical circle of radius R with angular velocity 'ω'. When the sphere is at lowest point of its path, the elongation of the wire is `underline((4"ML"("R"omega^2+g))/(pi"D"^2"Y"))`.

Explanation:

Force at the lowest point

`"F"="Mg"+"mv"^2/"R"`

F = Mg + MRω²

`"Y"="FL"/("A"Delta"L")=("M"("g"+"R"omega^2)"L")/((pi"D"^2)/4Deltal)`

`Deltal=(4"M"("g"+"R"omega^2)"L")/(pi"D"^2"Y")`