Question

AB is a vertical rigid infinite wire carrying a linear charge of density λ = 10 µC/m. A particle having mass m = 2 g and charge Q = 1 µC is fixed to the wire by means of a light, insulating and inextensible string having length = 2`sqrt2` m. The vertical velocity u in with which it should be projected under gravity from the shown position so that the string slacks when its angle with vertical becomes 45° is ______ (correct upto the first two decimal places).

Options

• 1.69

• 2.69

• 4.69

• 5.69

Answer 1

AB is a vertical rigid infinite wire carrying a linear charge of density λ = 10 µC/m. A particle having mass m = 2 g and charge Q = 1 µC is fixed to the wire by means of a light, insulating and inextensible string having length = 2`sqrt2` m. The vertical velocity u in with which it should be projected under gravity from the shown position so that the string slacks when its angle with vertical becomes 45° is 5.69 (correct upto the first two decimal places).

Explanation: 

Equation of circular motion

mg cos θ - `(2"k"lambda "Q"sintheta)/(ℓsintheta) = ("mv"^2)/ℓ`    ...(1)

Using work energy principle

– mgℓ cos θ + `int_ℓ^(ℓsintheta) (2"k"lambda "Qd"x)/x = 1/2`m(v² - u²)     ...(2)

By using (1) and (2)

u = 5.69 m/s