Question

A vector field F

is termed "conservative" because it conserves the total energy of a mass m in motion within this field. The total energy is defined as:
E=K+U=1/2m∣v∣2+U
where K represents kinetic energy, and U is the potential energy associated with F.
a) Consider a scenario in which a mass m is following a smooth curve γ(t) within a domain with a conservative force field F. Demonstrate that the total energy along the path, denoted as E(γ(t)), remains constant if the following relation holds:
F=mγ ′′(t). This is a manifestation of Newton's second law, f=ma, where a represents acceleration, a concept familiar from high school physics.
b) Show that the work performed by a conservative vector field F to transport to mass from point p0 to p1
is equal to: W=K(p1)−K(p0), where K(p) denotes the kinetic energy of the mass at point p.

Answer 1

A conservative force is one for which the work done is independent of path. In the scenario described, the force field F is conservative, meaning the work done by this force depends only on the starting and ending points of the motion, not on the path taken. The total energy along the path remains constant if the force field is conservative.

Step-by-step explanation:

A conservative force is one for which the work done is independent of path. In the scenario described, the force field F is conservative, meaning the work done by this force depends only on the starting and ending points of the motion, not on the path taken.

In order to demonstrate that the total energy along the path remains constant, we can use Newton's second law, which states that force is equal to mass times acceleration.

By rearranging the equation, we can find that the force (F) is equal to the mass (m) times the second derivative of the position vector (γ''). Therefore, if the force field is conservative, the total energy (E) along the path can be calculated as the sum of the kinetic energy (K) and the potential energy (U).

Since the work done by a conservative force is path independent, the change in potential energy will be equal to the negative of the change in kinetic energy, resulting in a constant total energy along the path.

To show that the work performed by a conservative vector field F to transport a mass from point p0 to p1 is equal to the difference in kinetic energy at those points, we can use the relationship between work and energy.

The work done by a force is equal to the negative change in potential energy. Therefore, for a conservative force field, the work done will be equal to the negative change in potential energy, which is equivalent to the change in kinetic energy