Question

In: Physics

Determine if the following forces are conservative a.) ? = ? 2??̂+ ? 2??̂+ ????̂ b)...

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.

Solutions

Expert Solution

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.


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