Dynamics: Conservation of energy

In Physics, a quantity is conserved when it does not change with time. Its value at some initial instant is the same as its value at some final instant, and at all times in between. If energy is conserved for a system, the total amount of energy remains the same, although if it changes in type of form. In the first chapter, we develop the principle of conservation of energy in one dimension and generalise it after.

1.Conservation energy: one dimension

1.2. Type of forces doing work on an object

1.2.1. Conservative forces

A conservative force is a force that does no net work on an object when this object does a round-trip motion, that is the starting point is the same as the ending point. The example is the one of the gravitational force.

An object moving up with a displacement + Δr undergoes a work ΔW_up = F_e . Δr done by the gravitational force F_e.

This work is negative, because F_e is negative and Δr is positive. The displacement is opposite the the force.

When returning downward to its starting point, it undergoes the work ΔW_down = + F_e . Δr

This work is positive, because F_e is negative and Δr is negative. The displacement and the the force have the same direction.

Therefore the net work done on this object by the gravitational force is:
W^e = W_up + W_down = - F_e . Δr + F_e . Δr = 0

Or by another way:
W = - mg (y^f - yⁱ) = - mg (yⁱ - yⁱ) = 0

The gravitational force F_e is conservative.

The gravitational force F_e is conservative

The second example is the force of a spring. We have seen that the work done on an object by a spring is: W = -(1/2) k (x_f² - x_f²) Then for a round-trip motion, we have W = -(1/2) k (x_i² - x_f²) = 0 The force of an elastic spring is conservative.

The force of a spring is conservative.

1.2.2. Nonconservative forces

In contrast, the work done by the kinetic frictional force depends on the direction of the object's motion, that is on the way the objects moves. This force does not depend on where the object is as the elastic spring or gravitational force do.

On a stationary surface, if an abject slides, the direction of the frictional force is always opposite to the velocity of the object. This force does a negative work for both parts of the round-trip motion. The dot product F_friction . Δr is always negative. The net work cannot be zero. It is equal for a round-trip to:
W_net = - 2 F_friction.Δr.
Therefore, the frictional force in nonconservative.

1.2.3. Conservative system

The forces doing work on the object are the forces due to the object or by its surroundings. A conservative system is the system in which only conservative forces do work. Hence, the conservation energy is valid only in a conservative system. An object moving on the floor with friction does not form a conservative system. An object falling on the ground with non negligible air-resistance effects in not a conservative system either.

2. Conservation energy law

Consider the case of a free-fall motion of an object, in which we neglect air-resistance effects. Then only the conservative force of gravitation does work on the falling object: W_net = - mg(y_f - y_i) We have then a conservative system. According to the work energy theorem, the work done on the object is equal to the change in kinetic energy of the object, so:

W_net = (1/2) mv_f² - (1/2)m v_i² , or
- mg (y_f - y_i) = (1/2) mv_f² - (1/2)m v_i²

Rearranging related terms gives:
mg y_i + (1/2)m v_i² = mg y_f + (1/2) mv_f²

Let's write:
mg y_i = U_i , the gravitational potential energy of the object at the position y_i, and
U_f at the position y_f,

(1/2)m v_i² = K_i the kinetic energy of the object at the position y_i, and
(1/2)m v_f² = K_f at the position y_f

Therefore:
U_i + K_i = U_f + K_f

U_i + K_i = U_f + K_f

The work energy theorem becomes the conservation of energy.

The above equality express the conservation of the quantity U(potential) + K(kinetic)

The gravitational potential U = mgy energy depends on the position of the moving object, and the kinetic energy K = (1/2)m v² depends on the speed of the moving object. Each of these energies changes during the motion of the object, however, their sum U + K = mgy + (1/2)m v² will remains constant throughout the motion of the object.

The sum of the kinetic energy K and the potential energy U is defined as the mechanical energyE .

During the motion, kinetic and potential energies are transformed into each other, such that the mechanical energy remains unchanged.

We can then write the conservation of energy as:
E_i = E_f

E_i = E_f

3. Net work done and potential energy

Let's recap:

Force conservative:
W_net (round-trip) = 0

Work done by a force:
W = ∫ F . dr
from r_i to r_f

Work-energy theorem:
W_net = (1/2) mv_f² - (1/2)m v_i² = K_f - K_i

Examples:
1.
Elastic spring force
F = - k x
Work done
W = - (1/2) k (x_f² - x_i²)

2.
Gravitational force
F = - mg
Work done
W = - mg (x_f - x_i )

Application:

Gravitational force and free-fall motion:

a. Work-energy theorem:

- mg (x_f - x_i) = (1/2) mv_f²- (1/2)m v_i²

b. Conservation of energy:

The latter equation leads to:
(1/2)m v_i² + mg x_i = (1/2) mv_f² + mg x_f, that is:
K_i + U_i = K_f + U_f
or
U_i - U_f = K_f - K_i = W_net
Therefore:

W_net = K_f - K_i = - ( U_f - U_i)
U_f - U_i = - W_net

The change in gravitational potential energy is equal to the negative work done by the conservative gravitational force.

We define then the potential energy for any conservative force as follows: The work done by a conservative force F(r) is:
W_net = ∫ F(r) dr
from r_i to r_f

Therefore:
The potential energy is:
U_f - U_i = - W_net= - ∫ F(r) dr

1. Gravitational potential energy

For a moving object under gravity, from y_i to y_f, the change in gravitational potential energy is:
U_f - U_i = - W_net = mg y_f - mg y_i

An increase in kinetic energy corresponds to a decrease in potential energy and vise versa.

It is important to notice that only the differences in potential energy are relevant. The orgin, that is the zero potential energy can be set at any point of our choice, called the refrence point. Usually, we select the reference, or level for gravitational potential energy at at y = 0, that is U = 0 at y = 0. Having made this choice, we have:
U(x) - U(0) = mg y - mg 0 , that is
U(y) = mgy

The expression of the potential energy depends on the choice of the origin, but changes in potential energy do not.

2. Elastic potential energy

For a spring of costant k, streched oe compressed from x_i to x_f, the change in elestic potential energy is:
U_f - U_i = - W_net = (1/2) k x_f² - (1/2) k x_i²

An increase in kinetic energy corresponds to a decrease in potential energy and vise versa.

Let's select the refrence point at x = 0, that is U = 0 at x = 0, the position of equilibrium when the spring is relaxed, neither streched not compressed. Having made this choice, we have:
U(x) - U(0) = (1/2) k x_f² - (1/2) k 0 ² , that is
U(x) = (1/2) k x²

The expression of the potential energy depends on the choice of the origin, but changes in potential energy do not.