MECHANICAL ENERGY: FORMULAS, CONCEPT, TYPES, EXAMPLES, EXERCISES - PHYSICAL - 2024

The mechanical energy of an object or a system is defined as the sum of its potential energy and its kinetic energy. As its name indicates, the system acquires mechanical energy thanks to the action of mechanical forces such as weight and elastic force.

Depending on the amount of mechanical energy the body has, it will also have the ability to perform mechanical work.

Figure 1. The motion of the roller coaster car can be described by the conservation of mechanical energy. Source: Pixabay.

Energy - of whatever type - is a scalar quantity, therefore lacking direction and meaning. Let E _m the mechanical energy of an object, U its potential energy and K its kinetic energy, the formula to calculate it is:

The unit in the International System for energy of any type is the joule, which is abbreviated as J. 1 J equals 1 Nm (newton per meter).

Regarding kinetic energy, it is calculated as follows:

Where m is the mass of the object and v its speed. Kinetic energy is always a positive quantity, since mass and the square of the velocity are. Regarding potential energy, if it is gravitational potential energy, we have:

Here m is still the mass, g is the acceleration of gravity and h is the height with respect to the reference level or if you prefer, the ground.

Now, if the body in question has elastic potential energy - it could be a spring - it is because it is compressed or perhaps elongated. In that case the associated potential energy is:

With k as the spring constant, which indicates how easy or difficult it is to deform and x the length of said deformation.

Concept and characteristics of mechanical energy

Going deeper into the definition given before, the mechanical energy then depends on the energy associated with the movement of the body: the kinetic energy, plus the contribution of the potential energy, which as we have already said can be gravitational, due to both its weight and the position of the body with respect to the ground or reference level.

Let's illustrate this with a simple example: suppose you have a pot on the ground and at rest. Since it is still, it has no kinetic energy, and it is also on the ground, a place from which it cannot fall; therefore it lacks gravitational potential energy and its mechanical energy is 0.

Now suppose someone places the pot right on the edge of a roof or window, 3.0 meters high. For this the person had to do work against gravity. The pot now has gravitational potential energy, it can fall from that height and its mechanical energy is no longer zero.

Figure 2. A flowerpot at the top of a window has gravitational potential energy. Source: Pixabay.

In these circumstances the pot has E _m = U and this amount depends on the height and weight of the pot, as indicated before.

Let's say the pot falls over because it was in a precarious position. As it falls, its speed increases and with it its kinetic energy, while the gravitational potential energy decreases, because it loses height. The mechanical energy at any instant of the fall is:

Conservative and non-conservative forces

When the pot is at a certain height, it has gravitational potential energy because whoever raised it did in turn work against gravity. The magnitude of this work is equal to that of gravity when the pot falls from the same height, but it has the opposite sign, since it was done against it.

The work done by forces such as gravity and elasticity only depends on the initial position and the final position that the object acquires. The path followed to go from one to the other does not matter, only the values ​​themselves matter. Forces that behave in this way are called conservative forces.

And because they are conservative, they allow the work done by them to be stored as potential energy in the configuration of the object or system. That is why the pot on the edge of the window or the roof, had the possibility of falling, and with it to develop movement.

Instead there are forces whose work depends on the path followed by the object on which they act. Friction belongs to this type of force. The soles of your shoes will wear more when you go from one place to another on a road with many turns, than when you go by a more direct one.

Friction forces do work that lowers the kinetic energy of bodies, because it slows them down. And that is why the mechanical energy of the systems in which friction acts tends to decrease.

Some work done by force is lost by heat or sound, for example.

Types of mechanical energy

Mechanical energy is, as we said, the sum of kinetic energy and potential energy. Now, potential energy can come from various conservative forces: weight, elastic force and electrostatic force.

- Kinetic energy

Kinetic energy is a scalar quantity that always comes from motion. Any particle or object in motion has kinetic energy. An object moving in a straight line has translational kinetic energy. The same happens if it is rotating, in which case we speak of rotational kinetic energy.

For example, a car traveling on a road has kinetic energy. Also a soccer ball while moving around the field or the person hurrying to get to the office.

- Potential energy

It is always possible to associate with a conservative force a scalar function called potential energy. The following are distinguished:

Gravitational potential energy

The one that all objects have by virtue of their height from the ground, or the reference level that has been selected as such. As an example, someone who is at rest on the terrace of a 10-story building has 0 potential energy with respect to the terrace floor, but not with respect to the street that is 10 floors below.

Elastic potential energy

It is usually stored in objects such as rubber bands and springs, associated with the deformation they experience when stretched or compressed.

Electrostatic potential energy

It is stored in a system of electric charges in equilibrium, due to the electrostatic interaction between them. Suppose we have two electric charges of the same sign separated by a small distance; since electric charges of the same sign repel each other, it is to be expected that some external agent has done work to bring them closer together.

Once they are positioned, the system manages to store the work that the agent did to configure them, in the form of electrostatic potential energy.

Conservation of mechanical energy

Returning to the falling pot, the gravitational potential energy that it had when it was on the edge of the roof is transformed into kinetic energy of motion. This increases at the expense of the first, but the sum of both remains constant, since the fall of the pot is activated by gravity, which is a conservative force.

There is an exchange between one type of energy and another, but the original amount is the same. Therefore it is valid to affirm that:

Alternatively:

In other words, the mechanical energy does not change and ∆E _m = 0. The symbol "∆" means variation or difference between a final and an initial quantity.

To correctly apply the principle of conservation of mechanical energy to problem solving, it is necessary to note that:

-It is only applied when the forces acting on the system are conservative (gravity, elastic and electrostatic). In this case: ∆E _m = 0.

-The system under study must be isolated. There is no energy transfer in any sense.

-If friction appears in a problem, then ∆E _m ≠ 0. Even so, the problem could be solved by finding the work done by the conservative forces, since it is the cause of the decrease in mechanical energy.

Deduction of the conservation of mechanical energy

Suppose that a conservative force acts on the system that does work W. This work causes a change in kinetic energy:

Equating these equations, since they both refer to the work done on the object:

The subscripts symbolize "final" and "initial". Grouping:

Examples of mechanical energy

Many objects have complex motions, in which it is difficult to find expressions for position, velocity, and acceleration as a function of time. In such cases, applying the principle of conservation of mechanical energy is a more efficient procedure than trying to apply Newton's laws directly.

Let's see some examples in which mechanical energy is conserved:

- A skier sliding downhill on snowy hills, provided frictionlessness is assumed. In this case, the weight is the force causing the movement along the entire trajectory.

- Roller coaster carts are one of the most typical examples. Here, too, weight is the force that defines movement and mechanical energy is conserved if there is no friction.

- The simple pendulum consists of a mass attached to an inextensible cord –the length does not change-, which is briefly separated from the vertical and is allowed to oscillate. We know that it will eventually brake from friction, but when friction is not considered, mechanical energy is also conserved.

- A block that impacts a spring fixed at one end to the wall, all placed on a very smooth table. The block compresses the spring, travels a certain distance, and is then thrown in the opposite direction because the spring is stretched. Here the block acquires its potential energy thanks to the work that the spring does on it.

- Spring and ball: when a spring is compressed by a ball, it bounces. This is because when the spring is released, the potential energy is converted to kinetic energy in the ball.

- Trampoline jump: it works in a similar way to a spring, elastically propelling the person who jumps on it. This makes use of its weight when jumping, with which it deforms the springboard, but this, when returning to its original position, provides the impetus to the jumper.

Figure 3. The trampoline acts like a spring, propelling people who jump on it upward. Source: Pixabay.

Solved exercises

- Exercise 1

An object of mass m = 1 kg is dropped down a ramp from a height of 1 m. If the ramp is extremely smooth, find the velocity of the body just as the spring collides.

Figure 4. An object descends on a ramp without friction and compresses a spring that is attached to the wall. Source: F. Zapata.

Solution

The statement informs that the ramp is smooth, which means that the only force acting on the body is its weight, a conservative force. Thus, it is indicated to apply the conservation of mechanical energy between any points of the path.

Consider the points marked in figure 5: A, B and C.

Figure 5. The path that the object follows is friction-free and mechanical energy is conserved between any pair of points. Source: F. Zapata.

It is possible to set conservation of energy between A and B, B and C or A and C, or any of the points in between on the ramp. For example, between A and C you have:

As it is released from point A, the velocity v _A = 0, on the other hand h _C = 0. Furthermore, the mass m cancels, as it is a common factor. So:

- Exercise 2

Find the maximum compression that the spring in exercise 1 will experience, if its elastic constant is 200 N / m.

Solution

The spring constant of the spring indicates the force that needs to be applied to deform it by one unit of length. Since the constant of this spring is k = 200 N / m, this indicates that 200 N are required to compress or stretch it 1 m.

Let x be the distance that the object compresses the spring before stopping at point D:

Figure 6. The object compresses the spring a distance x and stops momentarily. Source: F. Zapata.

The conservation of energy between points C and D, establishes that:

At point C it has no gravitational potential energy, since its height is 0, but it has kinetic energy. D has stopped completely, so there for K _D = 0, but instead makes available the potential energy of the compressed spring U _D.

The conservation of mechanical energy is as:

½ mv _C ² = ½ kx ²

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Figueroa, D. 2005. Series: Physics for Sciences and Engineering. Volume 1. Kinematics. Edited by Douglas Figueroa (USB).
3. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
4. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1.
5. Wikipedia. Mechanical energy Recovered from: es.wikipedia.org.