MECHANICAL POWER: WHAT IS IT, APPLICATIONS, EXAMPLES - PHYSICAL - 2025

The mechanical power is the rate at which the work, which is expressed mathematically by the amount of work done per unit of time is performed. And since work is done at the expense of absorbed energy, it can also be stated as energy per unit time.

Calling P to power, W to work, E to energy and t to time, all of the above can be summarized in easy-to-use mathematical expressions:

Figure 1. The Gossamer Albatross, the 'flying bicycle', crossed the English Channel in the late 1970s, using only human power. Source: Wikimedia Commons. Gossamer Albatross. Guroadrunner at English Wikipedia

O well:

It was named in honor of the Scottish engineer James Watt (1736-1819), known for creating the condenser steam engine, an invention that started the Industrial Revolution.

Other power units that are used in the industries are the hp (horse-power or horse power) and the CV (horsepower). The origin of these units also dates back to James Watt and the Industrial Revolution, when the measurement standard was the rate at which a horse did work.

Both hp and CV are roughly equal to ¾ kilo-W and are still widely used, especially in mechanical engineering, for example in the designation of motors.

Multiples of the watt, such as the aforementioned kilo-W = 1000 W are also often used in electrical power. This is because the joule is a relatively small unit of energy. The British system uses pound-feet / second.

What it consists of and applications in industry and energy

The concept of power is applicable to all types of energy, be it mechanical, electrical, chemical, wind, sonic or of any kind. Time is very important in industry, because processes must run as quickly as possible.

Any motor will do the necessary work as long as it has enough time, but the important thing is to do it in the shortest possible time, to increase efficiency.

A very simple application is immediately described to clarify well the distinction between work and power.

Suppose a heavy object is pulled by a rope. To do this, an external agent is required to do the necessary work. Let's say that this agent transfers 90 J of energy to the object-string system, so that it moves for 10 seconds.

In such a case, the energy transfer rate is 90 J / 10 s or 9 J / s. Then we can affirm that that agent, a person or a motor, has an output power of 9 W.

If another external agent is able to achieve the same displacement, either in less time or by transferring less energy, then it is capable of developing greater power.

Another example: suppose an energy transfer of 90 J, which manages to set the system in motion for 4 seconds. The output power will be 22.5 W.

Performance of a machine

Power is closely related to performance. The energy supplied to a machine is never completely transformed into useful work. An important part is usually dissipated in heat, which depends on many factors, for example the design of the machine.

That is why it is important to know the performance of the machines, which is defined as the quotient between the work delivered and the energy supplied:

Where the Greek letter η denotes the yield, a dimensionless quantity that is always less than 1. If it is also multiplied by 100, we have the yield in percentage terms.

Examples

- Humans and animals develop power during locomotion. For example, climbing stairs requires work against gravity. Comparing two people climbing a ladder, the one who climbs all the steps first will have developed more power than the other, but they both did the same job.

- Household appliances and machinery have their output power specified. An incandescent bulb suitable for lighting a room well has a power of 100 W. This means that the bulb transforms electrical energy into light and heat (most of it) at a rate of 100 J / s.

- The motor of a lawn mower can consume about 250 W and that of a car is on the order of 70 kW.

- A homemade water pump usually supplies 0.5 hp.

- The sun generates 3.6 x 10 ²⁶ W of power.

Power and speed

Instantaneous power is obtained by taking an infinitesimal time: P = dW / dt. The force that produces the work causing the small infinitesimal displacement d x is F (both are vectors), therefore dW = F d x. Substituting everything in the expression for the power, it remains:

Human power

People are capable of generating power of around 1500 W or 2 horsepower, at least for a short time, such as lifting weights.

On average, the daily power output (8 hours) is 0.1 hp per person. Much of which is translated into heat, roughly the same amount generated by a 75W incandescent bulb.

An athlete in training can generate an average of 0.5 hp equivalent to 350 J / s approximately, by transforming chemical energy (glucose and fat) into mechanical energy.

Figure 2. An athlete develops an average power of 2 hp. Source: Pixabay.

When it comes to human power, it is generally preferred to measure in kilo-calories / hour, rather than watts. The necessary equivalence is:

A power of 0.5 hp sounds like a very small amount, and it is for many applications.

However, in 1979 a human-powered bicycle was created that could fly. Paul MacCready designed the Gossamer Albatross, which crossed the English Channel generating 190 W of average output (Figure 1).

Distribution of electrical energy

An important application is the distribution of electrical energy between users. The companies that supply electricity bill for the energy consumed, not the rate at which it is consumed. That's why those who read your bill carefully will find a very specific unit: the kilowatt-hour or kW-h.

However, when the name Watt is included in this unit it refers to energy and not power.

The kilowatt-hour is used to indicate the consumption of electrical energy, since the joule, as mentioned before, is a fairly small unit: 1 watt-hour or Wh is the work done in 1 hour by a power of 1 watt.

Therefore 1 kW-h is the work that is done in an hour working with a power of 1kW or 1000 W. Let's put the numbers to convert these amounts to joules:

It is estimated that a household can consume about 200 kW-hours per month.

Exercises

Exercise 1

A farmer uses a tractor to pull a bale of hay M = 150 kg over a 15 ° inclined plane and into the barn at a constant speed of 5.0 km / h. The coefficient of kinetic friction between the hay bale and the chute is 0.45. Find the power output of the tractor.

Solution

For this problem, you need to draw a free-body diagram for the bale of hay that rises on the incline. Let F be the force applied by the tractor to lift the bale, α = 15º is the angle of inclination.

In addition, the kinetic friction force f _friction that opposes the movement is involved, plus the normal N and the weight W (do not confuse the W of the weight with that of the work).

Figure 3. Isolated body diagram of hay bale. Source: F. Zapata.

Newton's second law offers the following equations:

Speed ​​and force have the same direction and sense, therefore:

It is required to transform the units of the velocity:

Substituting values, finally we get:

Exercise 2

The motor shown in the figure will lift the 2 kg block, starting from rest, with an acceleration of 2 m / s ² and in 2 seconds.

Figure 4. A motor lifts an object to a certain height, for which it is necessary to do work and develop power. Source: F. Zapata.

Calculate:

a) The height reached by the block in that time.

b) The power that the engine must develop to achieve this.

Solution

a) It is a uniformly varied rectilinear motion, therefore the corresponding equations will be used, with initial velocity 0. The height reached is given by:

b) To find the power developed by the motor, the equation can be used:

And since the force that is exerted on the block is through the tension in the string, which is constant in magnitude:

P = (ma).y / Δ t = 2 kg x 2 m / s ² x 4 m / 2 s = 8 W

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 2. Dynamics. Edited by Douglas Figueroa (USB).
2. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
3. Physics Libretexts. Power. Recovered from: phys.libretexts.org
4. The Physics Hypertext Book. Power. Recovered from: physics.info.
5. Work, energy and power. Retrieved from: ncert.nic.in