Mechanical energy

January 1st, 2012

Energy is one of the concepts in physics, for which there exists no definition, i.e., undefined concept. We can feel the energy released in the form of heat when a conductor passes a current, we can estimate the reserve of the potential energy of the ball above the Earth’s surface according to the magnitude of its deformation after impact on the surface, etc.

Yet the energy concept is an intuitive concept.

American physicist Richard Feynman said that if there was a man who would have given the definition of energy, humankind would have put a monument to him. Despite the fact that energy is an undefined concept it is a physical quantity that has the remarkable property that the change in energy can be measured in different processes (note: not the energy, but only a change!).

Due to this property in physics emit different types of energy: the internal energy of the body, the energy of the electric field, the magnetic field energy, and the mechanical energy.

In this section, we consider mechanical energy, its types: potential and kinetic energy, as well as one of the fundamental laws of nature –  the law of conservation of mechanical energy.

The work of a force

For the mathematical formulation of the law let’s consider such a thing as the work of the force. Let the material point (abbreviation: MP) performs the movement of the point with coordinate  $x_1$ in a position $x_2$ under the action of   $\vec F$, which makes an angle $latex \alpha$ with a displacement vector (fig.1).

Fig.1. The calculation of the work of a force.

Let’s perform the expansion force $\vec F$ into two components: the normal component and the tangential component. The normal component $\vec F_n$ is perpendicular to the x-axis and does not perform work on the displacement of the MP which is moving along the x-axis. The work on the displacement of the MP which is moving along the x-axis carries out the tangential component. $\vec F_x$. This work is defined as the product of the magnitude of the tangential component  and the magnitude of the displacement $\Delta x=x_2-x_1$:

$A=F_x\Delta x=F$ $\Delta xcos\alpha$        (1)

If the force $\vec F$ is not constant and depends on the displacement (eg, elastic force), it is necessary to divide the movement into elementary sections  $dx$ within which the tangential component of the $\vec F_x$ can be considered as constant. Then the elementary work $dA$ can be represented as:

$dA=F_xdx$                     (2)

Integrating the last expression we obtain  an integral representation of the work of the force:

A=$\int_{x_1}^{x_2}F_xdx$                                            (3)

This expression for the work of the force will allow us to obtain an expression for the potential energy and kinetic energy at the same time as the two types of the same energy – mechanical energy.

The potential energy of a body in the gravity  field

Let’s represent the integral (3) as the difference between two integrals:

$A=\int_{0}^{x_2}F_xdx-\int_{0}^{x_1}F_xdx$ (4)

Let’s introduce the notation: U = $A=\int_{0}^{x}F_xdx$ and let’s call a scalar magnitude U as the potential energy of the body.

Let’s use the obtained relations for the analysis of a body moving in the gravity field (Fig. 2).

Fig.2. The movement of a body under the influence of gravity.

Let the body falls under the influence of gravity from the point with coordinate  $x_1$ to the point with coordinate $x_2$. Then, according to (4), the work that performs the force of gravity on the body movement is equal to:

$A=\int_{0}^{x_2} (-mg)dx-\int_{0}^{x_1}(-mg)dx=mgx_1-mgx_2$ (5)

Or:

$A=$-(U2-U1)  (6)

It can be shown that equation (6) holds not only for a rectilinear trajectory but also for the trajectory of the arbitrary shape. Let’s consider an arbitrarily curved trajectory (Fig. 3).

Fig.3. Calculation of the work of the gravity force.

Let’s partition the path on a set of mutually perpendicular segments. When a sufficiently large number of n segments – so large that n tends to infinity – the trajectory constructed by a broken line is a good approximation to a curved path. Then the work of the gravity force on the displacement of the body moving along a curved path can be a good approximation to replace the work of the same strength along the broken line.

The work of the gravity force along the broken line is the sum of the work of the gravity force along the straight line segments parallel to the force of gravity and of the work of the gravity force on straight stretches perpendicular to the gravity force. The work of the gravity force on straight stretches perpendicular to the force of gravity is equal to zero in accordance with the formula of the work of a force. Consequently, the work of the force of gravity along the broken line is the sum of the work of the gravity force along the straight line segments parallel to the force of gravity. It is clear that this sum is equal to the work of the gravity force performing the displacement of the body moving along a straight path between the points  $x_1$ and $x_2$.

Therefore, the work of the gravity force does not depend on the shape of the body trajectory. The forces that have this remarkable property, are called conservative forces.

Thus, the conservative forces have a place the equality (6).

If the work of the force depends on the shape of the trajectory, the force is called the dissipative force. An example of dissipative force can be friction force.

From the definition of work it is consequent that the work of the force is a positive value, therefore in the formula (6), U2<U1. Consequently, the potential energy of a body falling under gravity decreases. Where does disappear this energy? The answer to this question is given in the following section:

The kinetic energy of the moving body

Let’s consider the expression $dA=F_xdx$ and perform a series of simple transformations

$dA=F_xdx=F_xvdt=\frac{d(mv)}{dt}vdt=d(mv)v=mvdv$ (7)

The expression (7) makes it possible to present the work in a different form:

$A=\int_{x_1}^{x_2} F_xdx=\int_{v_1}^{v_2}mvdv=\frac{mv_2^2}{2}-\frac{mv_1^2}{2}$ (8)

The scalar value $\frac{mv^2}{2}$  is called the kinetic energy. We see from expression (8) that the work of the force is equal to the increase in the kinetic energy. It should be noted here that this result was obtained without specifying the type of force.

The law of conservation of the mechanical energy

Comparing the right and left sides of equations (5) and (8) we see that the work of the gravity force leads to the loss of potential energy (5), and at the same time the work of the force (in this case, the force of gravity) contributes to the increase of the kinetic energy:

$mgx_1- mgx_2=\frac{mv_2^2}{2}-\frac{mv_1^2}{2}$ (9)

Let us transform this equation into the form:

$mgx_1+\frac{mv_1^2}{2}= mgx_2+\frac{mv_2^2}{2}$ (10)

From equation (10) it follows that at each point of the trajectory the sum of the kinetic and potential energy of the system of bodies (the body in the gravitational field and the Earth, creating the field) is constant! The body and Earth form a system of bodies, that is:

1. closed system because the bodies interact with each other only;
2. conservative system because conservative forces act on each other.
These definitions allow us to formulate the law of conservation of mechanical energy in a more general form:

the sum of the potential and kinetic energy of a closed conservative system is constant.

Task 1.  A uniform cylinder of radius 4 cm is rolling on a horizontal surface at a speed of $3 \frac{m}{s}$ and reaches the foundation inclined plane with an angle of $30^0$ to the horizontal. Determine which height rises the cylinder on an inclined plane.
Task 2. The hammer weighing 400kg falls on a fixed pile weighing 100 kg at a speed of $5 \frac{m}{s}$. Determine the efficiency of the hammer blow.
Task 3.  To stretch the spring by 5 cm, it is necessary to perform the work $A_1$. To further stretch the spring by $\Delta x$ cm, you need to perform additional work $A_2$. Find $\Delta x$, if $\frac{A_2}{A_1}=3$.