# Work, Power And Energy

## Basic Theoretical Information

### Mechanical work

Energy characteristics of motion are introduced on the basis of the concept of **mechanical work or work of force** . The work done by a constant force *F* is a physical quantity equal to the product of the modules of force and displacement multiplied by the cosine of the angle between the vectors of force *F* and displacement *S* :

Work is a scalar value. It can be either positive (0 ° ≤ *α* <90 °) or negative (90 ° < *α* ≤ 180 °). At *α* = 90 °, the work done by force is zero. In the SI system, work is measured in joules (J). The joule is equal to the work done by a force of 1 newton on moving 1 meter in the direction of the force.

If the force changes over time, then to find work, build a graph of the dependence of force on displacement and find the area of the figure under the schedule – this is the work:

An example of a force whose modulus depends on the coordinate (displacement) is the elastic force of a spring, which obeys Hooke’s law ( *F *_{opr} = *kx* ).

### Power

The work of force performed per unit of time is called **power** . The power *P* (sometimes denoted by the letter *N* ) is a physical quantity equal to the ratio of work *A* to the time interval *t* during which this work is done:

According to this formula, the **average power is** calculated , i.e. power characterizes the process. So, the work can be expressed through power: *A* = *Pt* (if of course the power and the time of the work is known). A unit of power is called a watt (W) or 1 joule per 1 second. If the motion is uniform, then:

According to this formula, we can calculate the **instantaneous power** (power at a given time), if instead of the speed we substitute the value of the instantaneous velocity into the formula. How to find out what power to count? If the problem asks for power at a time or at some point in space, then it is considered instantaneous. If you ask about the power for a certain period of time or a section of the path, then look for the average power.

**Efficiency – efficiency** , equal to the ratio of useful work to expended, or the useful power to expended:

What work is useful and what work is determined from the condition of a specific task by logical reasoning. For example, if a crane does work to lift a load to a certain height, then work on lifting the load will be useful (since the crane was created just for the sake of it), and spent work would be work done by an electric crane motor.

So, the useful and expended power does not have a strict definition, and they are found by logical reasoning. In each task, we ourselves must determine what was the goal of the work (useful work or power) in this task, and what was the mechanism or method of doing all the work (power or work expended).

In general, efficiency shows how effectively a mechanism transforms one kind of energy into another. If the power changes with time, then the work is found as the area of the figure below the graph of power versus time:

### Kinetic energy

A physical quantity equal to half the product of body mass per square of its velocity is called the **kinetic energy of the body (the energy of motion)** :

That is, if a car weighing 2000 kg moves at a speed of 10 m / s, then it has a kinetic energy equal to *Е *_{к} = 100 kJ and is capable of doing work at 100 kJ. This energy can be converted into heat (when braking a car, the rubber of the wheels, road and brake discs are heated) or it can be spent on the deformation of the car and the body that the car collided with (in an accident). When calculating the kinetic energy, it does not matter where the car is moving, since the energy, like the work, is a scalar quantity.

**The body has energy if it is able to do the work. **For example, a moving body has kinetic energy, i.e. energy of motion, and is able to do work on the deformation of bodies or giving acceleration to the bodies with which the collision will occur.

The physical meaning of kinetic energy: in order for a body at rest *m of* mass to move at a speed *v,* it is necessary to do work equal to the obtained value of kinetic energy. If a body of mass *m* moves with velocity *v* , then to stop it, it is necessary to do work equal to its initial kinetic energy. During braking, the kinetic energy is mainly (except in the case of a collision, when the energy goes to deformation) is “taken away” by friction force.

**The kinetic energy theorem: the work of the resultant force is equal to the change in the kinetic energy of the body:**

The kinetic energy theorem is also valid in the general case when the body moves under the action of a changing force, the direction of which does not coincide with the direction of movement. It is convenient to apply this theorem in problems of acceleration and deceleration of the body.

### Potential energy

Along with the kinetic energy or the energy of motion in physics, the concept of **potential energy or energy of interaction of bodies** plays an important role .

The potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the surface of the Earth). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of the body and is determined only by the initial and final positions (the so-called **conservative forces** ). The work of such forces on a closed trajectory is zero. This property has the power of gravity and the force of elasticity. For these forces, you can introduce the concept of potential energy.

**The potential energy of the body in the field of gravity of the Earth is** calculated by the formula:

The physical meaning of the body’s potential energy: potential energy is equal to the work that gravity does when the body is lowered to zero ( *h* is the distance from the center of gravity of the body to zero) If the body has potential energy, then it is able to do work when this body falls from height *h* to zero. The work of gravity is equal to the change in the potential energy of the body, taken with the opposite sign:

Often in the tasks on energy one has to find work on raising (turning over, getting out of the hole) the body. In all these cases, it is necessary to consider the movement not of the body itself, but only of its center of gravity.

The potential energy Ep depends on the choice of the zero level, that is, on the choice of the origin of the coordinates of the axis OY. In each task, the zero level is selected for reasons of convenience. Physical meaning is not the potential energy itself, but its change when the body moves from one position to another. This change does not depend on the choice of the zero level.

**The potential energy of the stretched spring is** calculated by the formula:

where: *k* – spring stiffness. A stretched (or compressed) spring can set in motion the body attached to it, that is, inform the body of kinetic energy. Therefore, such a spring has a reserve of energy. Stretching or compressing *x* must be calculated from the undeformed state of the body.

The potential energy of an elastically deformed body is equal to the work of the elastic force during the transition from a given state to a state with zero deformation. If in the initial state the spring was already deformed, and its elongation was equal to *x *_{1} , then upon transition to a new state with elongation *x *_{2} the elastic force will do work equal to the change in potential energy taken with the opposite sign (since the elastic force is always directed against the deformation body):

Potential energy during elastic deformation is the energy of interaction of separate parts of the body between themselves by the forces of elasticity.

The work of the friction force depends on the distance traveled (this kind of force, whose work depends on the trajectory and the distance traveled is called: **dissipative forces** ). The concept of potential energy for the friction force can not be entered.

### Efficiency

**Efficiency (Efficiency)** is a characteristic of the effectiveness of the system (device, machine) in relation to the conversion or transfer of energy. It is determined by the ratio of the useful energy used to the total amount of energy received by the system (the formula is already given above).

Efficiency can be calculated both through work and through power. Useful and expended work (power) is always determined by simple logical reasoning.

In electric motors, efficiency is the ratio of the (useful) mechanical work performed to the electrical energy received from the source. In heat engines – the ratio of useful mechanical work to the amount of heat expended. In electrical transformers, the ratio of electromagnetic energy produced in the secondary winding to the energy consumed by the primary winding.

By virtue of their commonality, the concept of efficiency makes it possible to compare and evaluate from a single point of view various systems such as nuclear reactors, electric generators and engines, thermal power plants, semiconductor devices, biological objects, etc.

**Due to inevitable energy losses due to friction, heating of surrounding bodies, etc. Efficiency is always less than one. **Accordingly, the efficiency is expressed in fractions of expended energy, that is, in the form of a correct fraction or in percent, and is a dimensionless quantity. Efficiency describes how a machine or mechanism works effectively. Efficiency of thermal power plants reaches 35–40%, internal combustion engines with supercharging and pre-cooling – 40–50%, dynamos and high-power generators – 95%, transformers – 98%.

The task in which you need to find efficiency or it is known, you need to start with a logical reasoning – what work is useful and what work is spent.

### The law of conservation of mechanical energy

**The** total **mechanical energy** is the sum of the kinetic energy (i.e., the energy of motion) and the potential (i.e., the energy of interaction of bodies by the forces of strength and elasticity):

If mechanical energy does not transform into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If the mechanical energy goes into heat, then the change in mechanical energy is equal to the work of the friction force or energy loss, or the amount of heat released, and so on, in other words, the change in total mechanical energy is equal to the work of external forces

The sum of the kinetic and potential energy of the bodies that make up the closed system (that is, that in which external forces do not act, and their work is equal to zero, respectively) and interacting with each other by the forces of elasticity and elasticity, remains unchanged:

This statement expresses the **law of energy conservation (LEC) in mechanical processes** . He is a consequence of Newton’s laws. The law of conservation of mechanical energy is executed only when the bodies in a closed system interact with each other by the forces of elasticity and aggression. In all problems on the law of conservation of energy there will always be at least two states of the system of bodies. The law states that the total energy of the first state will be equal to the total energy of the second state.

**Algorithm for solving problems on the law of energy conservation:**

- Find the points of the initial and final body position.
- Write down what or what energy the body possesses at these points.
- Equate the initial and final energy of the body.
- Add other necessary equations from previous topics in physics.
- Solve the resulting equation or system of equations by mathematical methods.

It is important to note that the law of conservation of mechanical energy made it possible to obtain a connection between the coordinates and velocities of the body at two different points of the trajectory without analyzing the law of body motion at all intermediate points. The application of the law of conservation of mechanical energy can greatly simplify the solution of many problems.

In real conditions, almost always moving bodies, along with forces of elasticity and other forces, are acted upon by friction forces or environmental resistance forces. The work force of friction depends on the length of the path.

If friction forces act between the bodies that make up the closed system, then the mechanical energy is not conserved. Part of the mechanical energy is converted into the internal energy of the body (heating). Thus, the energy as a whole (ie, not only mechanical) is saved in any case.

**With any physical interactions, energy does not arise and does not disappear. It only turns from one form to another. **This experimentally established fact expresses the fundamental law of nature – the **law of conservation and transformation of energy** .

One of the consequences of the law of conservation and transformation of energy is the assertion that it is impossible to create a perpetual mobile (perpetuum mobile) – a machine that could do work indefinitely without spending energy.

### Different tasks for work

If the task requires finding a mechanical work, then first select the method of its finding:

- Work can be found by the formula:
*A*=*FS*∙ cos*α*. Find the force doing the work and the amount of body movement under the action of this force in the chosen frame of reference. Note that the angle must be selected between the force and displacement vectors. - The work of external force can be found as the difference of mechanical energy in the final and initial situations. Mechanical energy is equal to the sum of the kinetic and potential energies of the body.
- Work on lifting the body at a constant speed can be found by the formula:
*A*=*mgh*, where*h*is the height to which the**center of gravity of the body**rises . - Work can be found as a product of power for a while, i.e. according to the formula:
*A*=*Pt*. - The work can be found as the area of the figure under the graph of force versus displacement or power versus time.

### The law of conservation of energy and the dynamics of rotational motion

The tasks of this topic are quite complex mathematically, but with the knowledge of the approach, they are solved using a completely standard algorithm. In all tasks you will have to consider the rotation of the body in a vertical plane. The decision will be reduced to the following sequence of actions:

- It is necessary to determine the point of interest to you (the point at which it is necessary to determine the speed of the body, the force of the thread tension, the weight, and so on).
- Write at this point the second law of Newton, given that the body rotates, that is, it has a centripetal acceleration.
- Write the law of conservation of mechanical energy so that it contains the speed of the body at that very interesting point, as well as the characteristics of the state of the body in some state about which something is known.
- Depending on the condition, express the velocity squared from one equation and substitute it into another.
- Perform the rest of the necessary mathematical operations to obtain the final result.

**When solving problems one must remember that:**

- The condition of passing the top point when rotating on the thread with the minimum speed is the reaction force of the support
*N*at the top point equal to 0. The same condition is satisfied when passing the top point of the dead loop. - When rotating on a rod, the condition for passing the whole circle: the minimum speed at the top point is 0.
- The condition for the body to detach from the surface of the sphere – the reaction force of the support at the separation point is zero.

### Inelastic collisions

The law of conservation of mechanical energy and the law of conservation of momentum allow us to find solutions to mechanical problems in cases where the acting forces are unknown. An example of such tasks is the shock interaction of bodies.

**Impact (or collision)** is called short-term interaction of bodies, as a result of which their speeds undergo significant changes. During the collision of bodies between them there are short-term shock forces, the magnitude of which, as a rule, is unknown. Therefore, one cannot consider shock interaction directly using Newton’s laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude from consideration the process of collision itself and to obtain a relationship between the velocities of bodies before and after a collision, bypassing all intermediate values of these quantities.

Impact interaction of bodies often has to be dealt with in everyday life, in engineering and in physics (especially in the physics of the atom and elementary particles). In mechanics, two models of shock interaction are often used – **absolutely elastic and absolutely inelastic impacts** .

**Absolutely inelastic shock** is called such a shock interaction, in which the bodies join (stick together) with each other and move on as one body.

With an absolutely inelastic impact, mechanical energy is not conserved. It partially or completely goes into the internal energy of the bodies (heating). To describe any blows, you need to write down both the law of conservation of momentum and the law of conservation of mechanical energy, taking into account the heat released (it is extremely desirable to draw a picture beforehand).

### Absolutely elastic blow

**Absolutely elastic impact** is called a collision, in which the mechanical energy of a system of bodies is conserved. In many cases, the collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact. With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is satisfied. A simple example of an absolutely elastic collision can be the central blow of two billiard balls, one of which was at rest before the collision.

**The central impact of the** balls is a collision, in which the speeds of the balls before and after the impact are directed along the line of the centers. Thus, using the laws of conservation of mechanical energy and momentum, one can determine the speeds of balls after a collision, if their velocities are known before the collision. The central strike is very rarely implemented in practice, especially when it comes to collisions of atoms or molecules. In the case of an off-center elastic collision, the velocities of the particles (balls) before and after the collision are not directed in one straight line.

A special case of non-central elastic impact can be collisions of two billiard balls of the same mass, one of which was fixed before the impact, and the speed of the second was not directed along the lines of the centers of the balls. In this case, the velocity vectors of the balls after the elastic collision are always directed perpendicular to each other.

### The laws of conservation. Challenging tasks

#### Several bodies

In some tasks on the law of conservation of energy, cables with the help of which certain objects move can have a mass (that is, not be weightless, as you might already get used to). In this case, the work of moving such cables (namely, their centers of gravity) must also be taken into account.

If two bodies connected by a weightless rod rotate in a vertical plane, then:

- choose a zero level for calculating potential energy, for example, at the level of the axis of rotation or at the level of the lowest point where one of the loads is located, and they must make a drawing;
- they write down the law of conservation of mechanical energy, in which the left side records the sum of the kinetic and potential energy of both bodies in the initial situation, and the right side records the sum of the kinetic and potential energy of both bodies in the final situation;
- they take into account that the angular velocities of the bodies are the same, then the linear velocities of the bodies are proportional to the radii of rotation;
- if necessary, write down Newton’s second law for each of the bodies separately.

#### Projectile rupture

In the event of a projectile rupture, the energy of explosives is released. To find this energy, it is necessary to take away the mechanical energy of the projectile before the explosion from the sum of the mechanical energies of the fragments after the explosion. We will also use the law of conservation of momentum, written down, in the form of a cosine theorem (vector method) or in the form of projections onto selected axes.

#### Heavy slab collisions

Let a light ball of mass *m* with speed *u *_{n} move towards a heavy plate that moves with speed *v* . Since the impulse of the ball is much less than the impulse of the plate, then after the impact the speed of the plate will not change, and it will continue to move at the same speed and in the same direction. As a result of the elastic impact, the ball will fly away from the plate. Here it is important to understand that the **speed of the ball relative to the plate will not change** . In this case, for the final ball speed we get:

Thus, the speed of the ball after impact increases by twice the speed of the wall. A similar reasoning for the case when the ball and the plate moved in the same direction before the impact leads to the result that the speed of the ball decreases by twice the speed of the wall:

#### Problems about the maximum and minimum values of the energy of colliding balls

In problems of this type it is important to understand that the potential energy of the elastic deformation of the balls is maximal, if the kinetic energy of their movement is minimal — this follows from the law of conservation of mechanical energy. The sum of the kinetic energies of the balls is minimal at the moment when the speeds of the balls will be the same in magnitude and directed in the same direction. At this moment, the relative velocity of the balls is zero, and the deformation and the potential energy associated with it is maximum.