Air resistance is a force with magnitude proportional to v^{2}, and always acts
in the opposite direction of the velocity of the particle. Is air resistance a
conservative force?

Yes. Consider an object thrown into the air, reaching a maximum height, then
returning to the ground, thus completing a round trip. By our first principle of
conservative forces, the total work done by air resistance over this closed loop
must be zero. However, since air resistance always opposes the motion of
objects, it acts in the opposite direction as the displacement of the object for
the entire trip. Thus the net work over the closed loop must be negative, and
air resistance, much like friction, is a nonconservative force.

Problem :

A small disk of mass 4 kg moves in a circle of radius 1 m on a horizontal
surface, with coefficient of kinetic friction of .25. How much work is done by
friction during the completion of one revolution?

As we know with frictional force, the force exerted on the disc is constant
throughout the journey, and has a value of F_{k} = μ_{k}F_{n} = (.25)(4kg)(9.8m/s^{2}) = 9.8N. At every point on the circle, this force points in the opposite
direction of the velocity of the disk. Also the total distance traveled by the
disc is x = 2Πr = 2Π meters. Thus the total work done is:
W = Fx cosθ = (9.8N)(2Π)(cos180^{o}) = - 61.6 Joules. Note that over
this closed loop the total work done by friction is nonzero, proving again that
friction is a nonconservative force.

Problem :

Consider the last problem, a small disk traveling in circle. In this case,
however, there is no friction and the centripetal force is provided by a string
tied to the center of the circle, and the disk. Is the force provided by the
string conservative?

To decide whether or not the force is conservative, we must prove one of our two
principles to be true. We know that, in the absence of other forces, the tension
in the rope will remain constant, causing uniform circular motion. Thus, in one
complete revolution (a closed loop) the final velocity will be the same as the
initial velocity. Thus, by the Work-Energy Theorem, since there is no change in
velocity, there is no net work done over the closed loop. This statement proves
that the tension is, in this case, indeed a conservative force.

Problem :

Consider a ball being thrown horizontally, bouncing against a wall, then
returning to its original position. Clearly gravity exerts a net downward force
on the ball during the entire trip. Defend the fact that gravity is a
conservative force against this fact.

It is true that there is a net downward force on the ball. However, if the ball
is thrown horizontally, this force is always perpendicular to the displacement
of the ball. Thus, since force and displacement are perpendicular, no net
work is done on the ball, even though there is a net force. The net work
over the closed loop is still zero, and gravity remains conservative.

Problem :

Calculus Based Problem Given that the force of a mass on a spring is
given by F_{s} = - kx, calculate the net work done by the spring over one complete
oscillation: from an initial displacement of d, to -d, then back to its original
displacement of d. In this way confirm the fact that the spring force is
conservative.

To calculate the total work done during the trip, we must evaluate the integral
W = F(x)dx. To since the mass changes directions, we must
actually evaluate two integrals: one from d to –d, and one from –d to d: