Question

A thin wire of uniform linear density λ is bent into a semi-circle of radius R. What is the intensity of gravitational field at center O?

(1) \(\frac{2GM}{R^2}j\)

(2) \(\frac{2GM}{\pi R^2}j\)

(3) \(\frac{\sqrt{2}GM}{R^2}j\)

(4) \(\frac{3GM}{R^2}j\)

Answer 1

 (2) \(\frac{2GM}{\pi R^2}j\)

Consider the semi-circle as made up of large number of point masses (identical). The magnitude of dI is same but direction is different. The total intensity I = ∫ dl

Resolve dI and dI' into rectangular components along x and y–axis. The x–components cancel and y– componant and up. 

There resultant intensity I is along y–axis. Also