Two wires `AC` and `BC` are tied at `C` of a small sphere of mass `5kg`, wheich revolves at a constant speed `v` in the horizontal speed `v` in the horizontal circle of radius `1.6m`. Find the minimum value of `v`

Options

A. `4ms^(-1)`
B. `2ms^(-1)`
C. `2.5ms^(-1)`
D. none of these

Answer

A

Explanation

From the force diagram shown
`T_1cos30^@+T_2cos45^@=mg`………i
`T_1sin30^@+T_2sin45^@=(mv^2)/rmg` …………..ii
From eqn i and ii it is clear that the velocity will be minimum if `T_2` is minimum (may be zero also)
`T_1/2+T_2/sqrt2=(mv^2)/r`.............iii
After solving eqn i and ii we get
`T_1=(mg-(mv^2)/r)/(((sqrt3-1)/2)`
But `T_1gt0, (mg-(mv^2)/r)/((sqrt3-1)/2)gt0`
`mggt(mv^(2))/r=gtvltsqrt(rg)`
`v_(max)=sqrt(rg)=sqrt(1.6xx10)=4m//s`