In: Physics
Determine if the following forces are conservative
a.) ? = ? 2??̂+ ? 2??̂+ ????̂
b) ? = ??????̂+ ?????̂+ ???̂
c) ? = ??̂+ ??̂+ ??̂
d) ? = (3? 2?? − 3?)?̂+ (? 3 ? − 3?)?̂+ (? 3? + 2?)?̂
e) ? = −?? where a is a constant.
The work done by a conservative force is independent of the path; in other words, a force is conservative if the work it does around any closed path is zero: (let us consider the force )
Stokes' theorem: Stokes' theorem relates a line integral over a closed curve to a surface integral. If a path C is the boundary of some surface S, then Stokes' theorem says that
In order to ensure that is a conservative vector field, the condition should be
let us consider,
(a) here, F1= 2xy, F2=2xy, F3 = xyz
Hence the force is not conservative.
(b) here, F1= x.sin y, F2=cos y, F3 = xy
Hence the force is not conservative.
(c) here, F1= y, F2=z, F3 = x
Hence the force is not conservative.
(d) here, F1= 6xyz-3y, F2=3xz-3x, F3 = 3xy+2z
Hence the force is not conservative.
(e) here, F1= -ax, F2=-ay, F3 = -az
Hence the force is conservative.