**Problem : **

A skier glides down a frictionless hill of 100 meters, the ascends another hill, of height 90 meters, as shown in the figure below. What is the speed of the skier when it reaches the top of the second hill?

The skier moves from point A to point B

The skier is in a conservative system, as the only force acting upon him is gravity. Instead of calculating the work done over the curved hills, we can construct an alternate path, because of the principle of path independence:

Theoretical Path of Skier

We can cancel the mass and solve for
*v*
_{f}
:

Thus the final velocity of the skier is 14 m/s.

**Problem : **

What was the change in potential energy in the last problem, given that the mass of the skier is 50 kg?

Remember that
*ΔU* = - *W*
. We had calculated that the gravitational force
exerted a work of
10*mg*
during the entire trip. Thus the change in potential
energy is simply the negative of this quantity:
*ΔU* = - 10*mg* = - 500*g* = - 4900
Joules. The potential energy lost is converted into kinetic energy, accounting
for the final velocity of the skier.

**Problem : **
What is the total energy of the mass-spring system shown below? The mass is
shown at its maximum displacement on the spring, 5 meters from the equilibrium
point.

Mass-spring system

Here we have a system of two conservative forces, mass and gravity. Even if
there are more than one conservative force acting in a system, it is still a
conservative system. Thus potential energy is defined, and we can calculate the
total energy of the system. Since this quantity is constant, we may choose any
position for the mass that we like. In order to avoid calculating kinetic
energy, we choose a point at which the mass has no velocity: at its maximum
displacement, the position shown in the figure above. Also, since energy is
relative, we may choose our origin to be the equilibrium point of the spring, as
shown in the figure. Thus both the gravitational force and the spring force
contribute to the potential energy:
*U*
_{G} = *mgh* = - 5*mg* = - 245
Joules. Also,
*U*
_{s} =
*kx*
^{2} = (10)(5)^{2} = 125
Joules. Thus the total potential
energy, and hence the total energy is the sum of these two quantities:
*E* = *U*
_{G} + *U*
_{s} = - 120
Joules. Remember that answers may vary on this problem. If we
had chosen a different origin for our calculations, we would have gotten a
different answer. Once we have chosen an origin, however, the answer for total
energy must remain constant.

**Problem : **

A particle, under the influence of a conservative force, completes a circular path. What can be said about the change in potential energy of the particle after this journey?

We know that if the particle completes a closed path, the net work on the
particle is zero. We already established through the Work-Energy Theorem that
the total kinetic energy does not change. However, we also know that
*ΔU* = - *W*
. Since no work is done, the potential energy of the system does not change.

We can also answer this question in a more conceptual manner. We have defined potential energy as the energy of configuration of a system. If our particle returns to its initial position, the configuration of the system is the same, and must have the same potential energy.

**Problem : **

A pendulum with string of length 1 m is raised to an angle of
30^{
o
}
below
the horizontal, as shown below, and then released. What is the velocity of the
pendulum when it reaches the bottom of its swing?

A pendulum, shown in its initial and final positions.

In this case there are two forces acting on the ball: gravity and tension from the spring. The tension, however, always acts perpendicular to the motion of the ball, thus contributing no work to the system. Thus the system is a conservative one, with the only work being done by gravity. When the pendulum is elevated, it has a potential energy, according to its height above its lowest position. We can calculate this height:

Pendulum, with important distances shown