In a certain region of space, the gravitational field is given by `-(k)/(r)` where `r` is the distance and k is a constant. If the gravitaional potential at `r=r_(0)` be `V_(0)`, then what is the expression for the gravitaional potential (V)-

Options

A. `klog((r)/(r_(0)))`
B. `k log((r_(0))/(r))`
C. `V_(0)+klog((r)/(r_(0)))`
D. `V_(0)+klog((r_(0))/(r))`

Answer

D

Explanation

`intDV=-intE.dr,intdV=-int(k)/(r)dr`
`v=-klogr+c` at `r=r_(0),v_(0)`
`impliesv_(0)=-klogr_(0)+cimpliesc=v_(0)+klogr_(0)`
By substituting the value c from equation
`v=klog((r_(0))/(r))+V_(0)`
`vecF=-(dU)/(dr)=(d)/(dr)(ax+6y),F_(x)=-a` and `F_(y)=b`
`vecF=-ahati+bhatjimplies"acceleration"=(sqrt(a^(2)+b^(2)))/(m)`
`Tsintheta=(Gm^(2))/(l^(2)),Tcostheta=mg`
`tantheta=(Gm)/(gl^(2)),theta=tan^(-1)((Gm)/(gl^(2)))`