A small sized mass `m` is attached by a massless string (of length `L`) to the top of a fixed frictionless solid cone whose axis is vertical. The half angle at the vertex of cone is theta. If the mass `m` moves around in a horizontal circle at speed `v` , what is the maximum value of `v` for which mass stay in contact with the comes? (g is acceleration due to gravity.)

Options

A. the car cannot make a turn without skidding
B. if the car runs at a speed less than `40km//hr`, it will slip up the slope
C. If the car runs at the correct speed of `40km//hr` , the force by the road on car is equal to `mv^(2)//r`
D. if the car runs at the correct speed of `40km//hr` , the force by the road on the car is greater than mg as well as greater than `mv^(2)//r`

Answer

D

Explanation

Car can make a turn without skidding at `40km//hr` . The car will slip down the slope if it runs at a speed less than `40km//hr` . `Nsintheta=(mv^(2))/(r)` implies `N=(1)/(sintheta)((mv^(2))/(r))`
and `Ncostheta=mg` implies `N=(mg)/(costheta)` :. `sintheta` and `costhetalt1`
So `Ngtmg` and `Ngt(mv^(2))/(r)`