A ball is attached to a string of length `L` and the ball is given velocity `v_0` at its lowest point. Find the speed of the ball at point B,C and D. Assume that the ball is moving in vertical circle.

Explanation

Here forces acting on the ball are gravitational force and tension in the string. Gravitational force is conservative and work done by tension is zero, because it is always perpendicular to displacement. Hence, conservation of mechanical energy is applicable.
Assuming a horizontal line through A as the reference line,

`K_A=(1)/(2)mv_0^2`,`U_A=0`
`K_B=(1)/(2)mv_1^2`,`U_B=mg=mgL(1-cosalpha)`
`K_A+U_A=K_B+U_B`
`(1)/(2)mv_0^2+0=(1)/(2)mv_1^2+mgL(1-cosalpha)`
`v_1=sqrt(v_0^2-2gL(1-cosalpha))`

`K_C=(1)/(2)mv_2^2`,`U_C=mg(L+Lcosbeta)`
`K_A+U_A=K_C+U_C`
`v_2=sqrt(v_0^2-2gL(1+cosbeta))`

`K_D=(1)/(2)mv_3^2`,`U_D=mgxx2L` ltbegt `K_A+U_A=K_D_U_D`
`v_3=sqrt(v_0^2-4gL)`