Question

A particle is moving on frictionless XY-plane. It is acted on by a conservative force described by the potential-energy function.
`U(x,y,z)=(1)/(2)k(x^(2)+y^(2)+z^(2))`
where , k=constant
Derive an expression for the force an the particle.

Answer 1

Force acting an the particle in x-direction is given by
`F_(x)=(delU)/(delx)=-(1)/(2)k(2x+0+0)=-kx`
Similarly, `F_(y)=-ky` and `F_(z)=-kz`
`:.` Force acting an the particle is
`F=Fxhati+Fyhatj+Fzhatk=-k(xhati+yhatj+zhatk)=-kx`,
where r =position vector of the particle
`=xhati+yhatj+zhatk`