Work And Energy

Unit: Energy

Chapter: Conservation of Energy, the work-energy principle, and Power

Reference: AP Physics Algebra, Energy, Conservation of Energy, the work-energy principle, and Power, Work done by a variable force, The Work-Energy Theorem for a Variable Force Various forms of energy, Law of Conservation of Energy, The concept of potential energy, The potential energy of springs, The conservation of mechanical energy, Power

After studying this chapter, you should be able to:

• calculate the work done by gravity when a mass moves from one point to another;
• explain the meaning of energy;
• obtain expressions for gravitational potential energy and elastic potential energy
• define the power of a system;

Work done by a variable force

You have so far studied the cases where the force acting on the object is constant. This may not always be true. In some cases, the force responsible for doing work may keep varying with time. Let us now consider a case in which the magnitude of force F(x) changes with the position x of the object. Let us now calculate the work done by a variable force. Let us assume that the displacement is from xi to x_f, where x_i and x_f are the initial and final positions. In such a situation, work is calculated over a large number of small intervals of displacements Δx. In fact, Δx is taken so small that the force F(x) can be assumed to be constant over each such interval. The work done during small displacements Δx is given by

ΔW = F(x) Δx

F(x) Δx is numerically equal to the small area shown shaded in Fig. given below. The total work done by the force between xi and xf is the sum of all such areas (area of all strips added together):

W = ΣΔW

= ΣF (x) Δx

A varying force F moves the object from the initial position xi to the final position x_f. The variation of force with distance is shown by the solid curve (arbitrary) and the work done is numerically equal to the shaded area.

The width of the strips can be made as small as possible so that the areas of all strips added together are equal to the total area enclosed between xi and x_f. It will give the total work done by the force between x_i and x_f:

The Work-Energy Theorem for a Variable Force

We are now familiar with the concepts of work and kinetic energy to prove the work-energy theorem for a variable force. We confine ourselves to one dimension. The time rate of change of kinetic energy

    = F/ v (from Newton’s Second Law)

                    =Fdx/dt

Thus,

dK = Fdx

Integrating from the initial position (x_i) to the final position (x_¦),

where K_i and K_¦ are the initial and final kinetic energies corresponding to x_i and x_¦.

or     

From

K_¦ − K_i = W                                                                       

Thus, the WE theorem is proved for a variable force.

Various forms of energy

Thermal Energy (Heat Energy)

 All matter is made up of particles that are constantly moving; therefore, all matter has kinetic energy.

1. At higher temperatures, particles move faster, thus having more kinetic energy and greater thermal energy.

2. Particles that are further apart have more energy than particles that are closer together.

3. Thermal energy also depends on the number of particles.

4. Ex: Steam has more energy than an ice cube and the ocean; but the ocean has the most thermal energy because it contains the most particles.

Chemical Energy

1. The energy of a compound changes as its atoms are arranged to form new compounds
2. Molecules that have a lot of bonds between atoms tend to have a lot of chemical energy- gasoline.
3.  Ex:

1. When wood burns, the chemical energy stored in the wood is used to heat the house.

2. When you eat a marshmallow, the chemical energy stored in the sugar molecules becomes available for you to use.

Electrical Energy

1. The energy of moving electrons
2.  The electrical energy produced by electrons moving (120 times per second) is used to do work.
3.  Generators rotate magnets within coils of wire to produce electrical energy.
4.  Electrical energy can be considered both potential energy (because the magnet is changing position) and kinetic energy (because the electrons are moving).

Sound Energy

1. Caused by an object’s vibration
2.  A form of potential and kinetic energy

To make an object vibrate, work must be done to change its position.

Ex: When you pluck and release a guitar string; when the guitar string moves back to its original position, it has kinetic energy.

Light Energy

1. Produced by the vibrations of electrically charged particles
2. Can be transmitted through a vacuum (a space without matter)

               The energy used to cook food in the microwave

Nuclear Energy (Atomic Energy)

The energy associated with changes in the nucleus of an atom is produced in 2 ways:

 1. When 2 or more nuclei join together

2. When the nucleus of an atom split apart

In the sun, hydrogen nuclei join together to make a larger helium nucleus. This reaction releases a huge amount of energy, which allows the sun to light and heat the Earth.

The nuclei of some atoms, such as Uranium, store a lot of potential energy. When work is done to split these nuclei apart, energy is released. This nuclear energy is used to generate electrical energy, which will run nuclear power plants.

Law of Conservation of Energy

(1) Law of conservation of energy: For an isolated system or body in the presence of conservative forces the sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend on time. This is known as the law of conservation of mechanical energy.

(2) Law of conservation of total energy: If the forces are conservative and non-conservative both, it is not the mechanical energy alone which is conserved, but it is the total energy, maybe heat, light, sound or mechanical etc., which is conserved.

The concept of potential energy:

In the previous section, we discussed that a moving object has kinetic energy associated with it. Objects possess another kind of energy due to their position in space. This energy is known as Potential Energy. A familiar example is the Gravitational Potential Energy possessed by a body in Gravitational Field. Let us understand it now

Potential Energy in Gravitational Field Supposes that a person lifts a mass m from a given height h₁ to a height h₂ above the earth’s surface. Let us also assume that the value of acceleration due to gravity remains constant. The mass has been displaced by a distance h = (h₂ – h₁ ) against the force of gravity. The magnitude of this force is mg and it acts downwards. Therefore, the work done by the person is

W = force × distance = mgh

The work is positive and is stored in mass m as energy. This energy by virtue of the position in

The object of mass m originally at height h₁ above the earth’s surface is moved to a height h₂.

space is called gravitational potential energy. It has the capacity to do work. If this mass is left free, it will fall down and during the fall it can be made to do work. For example, it can lift another mass if properly connected by a string, which is passing over a pulley. The selection of the initial height h₁ is arbitrary. The important concept is the change in height, i.e. (h₂ – h₁). We, therefore, say that the point of zero potential energy is arbitrary. Any point in space can be chosen as a point of zero potential energy. Normally, a point on the surface of the earth is assumed to be the reference point with zero potential energy.

= 2.56 × 10² = 256 hp.

The potential energy of springs:

You now know that an external force is required to compress or stretch a given spring. These situations are shown in Fig. (given below). Let there be a spring of force constant k. This spring is compressed by a distance x. From Eqn.12  kx²_m. We recall that work done by the external force to compress the spring is given by

W = 12  kx²

This work is stored in the spring as elastic potential energy. When the spring is left free, it bounces back and the elastic potential energy of the spring is converted into kinetic energy of the mass m.

The conservation of mechanical energy:

(a)Conservation of mechanical energy during the free fall of a body We now wish to test the validity of the law of conservation of energy in case of mechanical energy, which is of immediate interest. Let us suppose that an object of mass m lying on the ground is lifted to a height h. The work done is mgh, which is stored in the object as potential energy. This object is now allowed to fall freely. Let us calculate the energy of this object when it has fallen through a distance h₁. The height of the object now above the earth's surface is h₂ = h – h₁ (Fig given below). At this point P, the potential energy = mgh₂.

Mass m is lifted to a height h from the earth's surface. It is then lowered to a height h₂ at point P. The total energy at P is the same as that at the highest point.

When the object falls freely, it gets accelerated and gains in speed. We can calculate the speed of the object when it has fallen through a height h1 from the top positions using the equation

v² = u² + 2gs

where u is the initial speed at the height h₁, i.e., u = 0 and s = h₁.

Then, we have v² = 2gh₁

The kinetic energy at point P is given by K.E = 12  mv²

= m₂  × 2gh₁ = mgh₁

 The total energy at the point P is Kinetic Energy + Potential Energy

= mgh₁ + mgh₂ = mgh

This is the same as the potential energy at the highest point. Thus, the total Energy is conserved.

(b) Conservation of Mechanical Energy for a Mass Oscillating on a Spring Fig. given below. shows a spring whose one end is fixed to a rigid wall and the other end is connected to a wooden block lying on a smooth horizontal table. This free end is at x₀ in the relaxed position of the spring. A block of mass m moving with speed v along the line of the spring collides with the spring at the free end and compresses it by x_m. This is the maximum compression. At x_0, the total energy of the spring-mass system is 12  mv² It is the kinetic energy of the mass. The potential energy of the spring is zero. At the point of extreme compression, the potential energy of the spring is  12  kx²_m and the kinetic energy of the mass is zero. The total energy now is 12  kx²_m. Obviously, this means that

12  kx²_m =12  mv²

A block of mass m moving with velocity v on a horizontal surface collides with the spring. The maximum compression is x_m.

A block of mass m moving with velocity v on a horizontal surface collides with the spring. The maximum compression is x_m.

K.E + P.E (Before collision) = K.E. + P.E. (After collision)

i.e., the total energy is conserved.

Power:

The rate at which work is done is called power.

If ΔW work is done in time Δt, the average power is defined as

Average Power = Work done time taken

Mathematically, we can write

If the rate of doing work is not constant, this rate may vary. In such cases, we may define instantaneous power P

The definition of power helps us to determine the SI unit of power:

= joule/ second = watt Thus, the SI unit of power is watt. It is abbreviated as W.

The power of an agent doing work is 1W, if one joule of work is done by it in one second. The more common units of power are kilowatt (kW) and megawatt (MW).

1 kW = 10³ W, and 1 MW= 10⁶ W

You may have heard electricians discussing the power of a machine in terms of horsepower, abbreviated as hp. This unit of power was under the British system. It is a larger unit: 1hp = 746 W

 The unit of power is used to define a new unit of work (energy). One such unit of work is the kilowatt hour. This unit is commonly used in electrical measurement.

kilowatt. hour (kWh) = 1 kW × 1 hour = 10³ W × 3600 s

= 36,00,000 J = 3.6 × 10⁶ J Or 1 kW h = 3.6 MJ (mega joules)

 The electrical energy that is consumed in homes is measured in kilowatt-hour.

 In common man’s language: 1kW h = 1 Unit of electrical energy consumption.

Example 1:

A truck is loaded with sugar bags. The total mass of the load and the truck together is 100,000 kg. The truck moves on a winding path up a mountain to a height of 700 m in 1 hour. What average power must the engine produce to lift the material?

Solution:

W = mgh = (100,000 kg) × (9.8 m s^–2 × 700 m)

= 9.8× 7× 10⁷ J

= 68.6 × 10⁷ J

Time taken = 1 hour = 60 × 60 s = 3600 s

Average Power, P = W/t

= 1.91 × 10⁵ W

We know that 746 W = 1 hp

Key Points:

Work done by a constant force F is

W = F.d = Fd cosθ

Where θ  is the angle between F and d. The unit of work is the joule. Work is a

scalar quantity.

Work is numerically equal to the area under the F versus x graph.

Work done by elastic force obeying Hooke’s law is W =12 kx²

where k is the force constant of the elastic material (spring or wire). The sign of W is positive for the external force acting on the spring and negative for the restoring force offered by the spring. x is compression or elongation of the spring.

Law of Conservation of Energy

(1) Law of conservation of energy: For an isolated system or body in the presence of conservative forces the sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend on time. This is known as the law of conservation of mechanical energy.

(2) Law of conservation of total energy: If the forces are conservative and non-conservative both, it is not the mechanical energy alone which is conserved, but it is the total energy, maybe heat, light, sound or mechanical etc., which is conserved.

Conservation of Energy:

The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. This means that the total amount of energy in a closed system remains constant. This principle is one of the most fundamental laws of physics and has numerous applications in different fields, from mechanics to thermodynamics, electromagnetism, and quantum mechanics.

Work-Energy Principle:

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In other words, work is the transfer of energy from one system to another, and the energy transferred can be either potential or kinetic. The work-energy principle is a powerful tool for analysing the motion of objects and calculating their velocities and accelerations.

Power:

Power is the rate at which work is done or energy is transferred. It is defined as the amount of work done per unit of time. The SI unit of power is watts (W), which is equal to one joule per second (J/s). Power is an important concept in physics, engineering, and technology, as it provides a way to quantify the efficiency of machines and processes. It is also a critical parameter in the design and optimization of power systems, such as engines, generators, and electric grids.