Work, Energy and Power

ENERGY

       The terms force and energy were not always clearly defined. Before the mid-nineteenth century, they were often used interchangeably. However, progress in mechanics and thermal physics helped clarify these ideas and the distinction between them. In 1807, the English scientist Thomas Young (1773 – 1829) introduced the word energy to denote the quantity of work that a system can do. Later, the Scottish engineer and thermal physicist W. M. J. Rankine coined the terms potential energy and conservation of energy. As is often the case in science, this classification of terms and definitions led to greater insights and understanding of natural laws and their consequences. Today the principle of conservation of energy is part of the framework of physical theory. Our faith in this principle is based on years of our experience.   

       Energy is a vital part of our daily lives. The food we eat gives our bodies energy for movement; electrical energy lights our homes and streets; oil and gas propel our cars and keep us warm.  The use of energy is growing at an incredible rate. In the U.S. alone, energy demands have increased more than 250%  since 1940. The increase maybe even more rapid in the years ahead. The energy demands are growing more rapidly than population itself. In other words, the use of energy per person is also rising.. The increase is largely a result of the improvement in the standard of living. The switch from human power to machine power has increased productivity and provided us with more goods and leisure time.  Unfortunately, the use of labor-saving devices places serious demands on our available energy supplies.

       The wise use of existing energy resources is one answer to this problem. By avoiding unnecessary use of energy for heating, lighting, air conditioning and transportation, serious energy shortages can be prevented.

       Conservation of energy is only a partial answer, however; new energy sources must be found and developed in order to achieve and maintain a desirable living standard for the growing population of the world.

        Actually, there is no shortage of energy at all. The sun floods the earth with enough radiant energy everyday to supply the whole world’s needs many times over. In fact, it has been calculated that present worldwide demands could be met if we could completely convert into electrical power the solar energy falling on a plot of ground  near the equator at a mere 125 miles square ( 201.1 km square ). It has also been estimated that the Gulf Stream in the Atlantic Ocean transports enough warm water to generate the electric energy needs of the U. S. many times over.

        The shortage lies in our knowledge and our means to convert the ample energy supplies in and around our planet into usable forms.

        Many sources of energy on the earth come directly from energy radiated from the sun. Coal and oil, the fossil fuels, were formed from the sun millions of years ago. Hydroelectric power depends upon dammed-up rain water that was evaporated from the sea by solar energy. Winds and ocean currents, potential sources of additional energy, are caused by solar energy.

       Some energy sources cannot be attributed to the sun. Nuclear energy comes from  changing matter into energy.  Geothermal energy ( heat from the earth ) uses heat locked in rocks since the Earth was formed as a molten mass. The Earth’s rotation and the gravitational attraction between the Earth and the moon moves large masses of water on the earth. These movements of water are called tides. In some places such as the Rance river in France, tides are usefully harnessed.

       The amount of radiant energy we receive from the sun each day is limited. Also, the amount of fossil  fuel is limited. Research efforts to supply more energy have aimed at increasing our supply of solar energy, using our solar resources more efficiently, and increasing energy production from non-solar sources. 

       ENERGY is the capacity or ability to do work. 

POTENTIAL ENERGY is an energy due to its position or elevation.  A body is said to have potential energy if by virtue of its state or position is able to do work.   Water at the top of a hydraulic dam has energy due to its position.. As the water runs downhill through a turbine, the potential energy of the water is converted to

electrical energy.

ELASTIC POTENTIAL ENERGY is an energy possess by a compressed or stretched spring.

KINETIC ENERGY is an energy of motion. The kinetic energy possess by a moving body is defined as the energy possess by the body by virtue of its motion. 

     Sources of energy :       

         1. Sun  -- main source                                                 

         2. Wind                                      

         3. Water   

         4. Geothermal 

         5. Nuclear    

         6. Fossil fuels ( coal, oil natural gas) 

                                                 

    

 Forms of energy :

         1.  Chemical energy                          

         2.  Electric energy

         3.  Thermal / Heat energy (internal energy)

         4.  Mechanical energy

                          Forms :  a) kinetic and  b) potential energies 

         5.  Nuclear energy 

         6.  Light energy (solar/ Radiant  energy)  

         7.  Atomic / molecular energy 

WORK

       The word work means many different things to us in our daily lives. We say that we work when we sweep the yard, buy groceries or drive a car. We also work if we drag or push an object across the floor. How much work we do depends on how hard we push and how far we move the object. In a layman’s point of view, work is the expenditure on one’s stored up bodily energy.  In the physical sciences, work is more precise and restricted than in everyday usage. Work is defined as the product of force and the displacement through which the force acts as the object moves.

       Factors to be considered in measuring work :

          1.  There must be an applied force.

          2.  The force must act through a displacement, S. 

          3.  The force must have a component F_x parallel to the direction of the 
               displacement.  

       If an applied force is not along the direction of motion, we can resolve it into components parallel to and perpendicular to the displacement. Only the component of the force that is parallel to the displacement contributes to the work.   

UNITS OF WORK

  1.  Joule ( J ) – one joule is the work done by a force of one Newton in moving an object through a parallel distance of one meter.     

                        1 J = 1 N . m 

  2.  Erg – one erg is the work done by a force of one dyne in moving  an object through a distance of one centimeter.     

                      1 erg = 1 dyne . cm 

  3.  Foot – pound ( ft-lb) – one foot pound is the work done by a force of one pound in moving an object through a parallel distance of one foot. 

                                                                                               

            W  = FS cos θ , if the force and the displacement are oblique with each other

            W  = FS  , if the force and the displacement are in the same direction. 

     When a mass m is lifted to a height h, the force exerted is equal to the weight of the mass, the work done against gravity approaches the potential energy and  S = h.  

            E_p = mgh , potential energy [ Work done against gravity ]        

        

  If the mass is release from rest the speed of the mass is given by v = √ 2gh  ,  
     v² = 2gh  and  h = v²/ 2g.

The potential energy, E_p undergoes transformation to kinetic energy, E_k.  

       E_p =  E_k = mg ( v²/ 2g ) =  ½ mv² ,  kinetic energy

     E_p = ½ ky², elastic potential energy ; k = spring constant,  
                         y  = elongation or deformation

CONVERSION OF MASS TO ENERGY

       In his special theory of relativity ( 1905 ) Einstein concluded that mass and energy are interchangeable. The quantitative mass-energy relationship is given in his equation, 

                E = mc²  ,  where  m = mass   and   c = speed of light = 3  x  10⁸ m/ s.

    Mass is converted into energy in nuclear rectors and nuclear weapons. As well as in the sun and other stars.  For nuclear reactors using  U²³⁵ as fuel, about 1/1000 of the mass of each fissioning atom is converted into other forms of energy.  Although the fraction of our energy needs supplied by nuclear reactors  on earth is relatively small, it is increasing rapidly as energy from oil and natural gas becomes less plentiful and more expensive. 

       THERMAL ENERGY OR INTERNAL ENERGY is associated with the random kinetic energies of the atoms and molecules in the object.

  FOODS AND OTHER FUELS

        Many of the most common energy sources are chemical in nature such as food, gasoline and natural gas. The energy content in foods are given in units of kilocalories. ( 1 kcal = 4186 joule ). For foods and fuels the process by which stored chemical energy is released is by oxidation. In machines, the oxidation process produces thermal energy which is partially converted to work and other forms of energy. In animals, the oxidation process is complex which also results both in thermal energy and work being performed by the animal. If the animal consumes more food than it needs, it will convert the excess to fat, which is another form of chemical energy. The chemical energy stored in fats is used if and when there is a food deficit.

        Dieting to lose weight would mean reducing of the food energy intake. Exercise aids dieting partially because more food energy is converted to work. 

CALORIC CONTENT ( kcal/g ) OF COMMON FOODS AND FUELS

COMMON FOODS	kcal/g	Eggs	1.63	Sirloin, lean	1.66
Apples	0.58	Grapes	0.69	Sugar	4.00
Avocado	1.67	Ham, cooked	2.23	Tomato	0.22
Baby formula	0.67	Hamburger, lean	1.63	Tuna, in oil	1.97
Beans, kidney	1.18	Ice cream,  chocolate	2.22	Wine	0.85
Beer	0.42	Lard ( fat )	9.30	COMMON  FUELS
Butter	7.20	Lobster, raw	0.91	Coal	8.00
Carrots	0.42	Milk, whole	0.64	Gasoline	11.4
Celery	0.14	Milk, low-fat	0.42	Furnace oil	10.5
Cheese, cheddar	4.00	Oranges	0.49	Methanol	5.20
Cheese, cottage	1.06	Peanuts, roasted	5.73	Natural gas	13.00
Chicken, roasted	1.60	Peas	0.71	Wood ( average )	4.00
Chocolate	5.28	Potato, baked	0.93
Coffee, black	0.008	Raisins	2.90	Average carbohydrates	4.10
Cola, carbonated	0.36	Rice, white, cooked	1.09	Average protein	4.10
Corn flakes	3.93	Shrimps, snails, raw	0.91	Average fat	9.30

ENERGY  CONSUMPTION  RATE for VARIOUS ACTIVITIES

ACTIVITY	RATE (kcal/ min)	Playing tennis	6.30
Sleeping	1.20	Swimming breaststroke	6.80
Sitting at rest	1.70	Ice skating (14.5 km/hr)	7.80
Standing, relaxed	1.80	Climbing stairs  (116/min)	9.80
Sitting in class	3.00	Cycling ( 21 km / hr )	10.00
Walking slowly ( 4.8 km/hr)	3.80	Playing basketball	11.40
Cycling ( 13–18 km/ hr)	5.70	Cycling, professional racer	26.50

  

ELECTRIC ENERGY

       Capacitor is a device which stores pure electric energy. Many electronic instruments,  such as the heart defibrillators use capacitors to store energy. Fibrillation is a potentially fatal malfunction of the beating of the heart. The electric energy stored in the large capacitor of the defibrillator is used to cause an electric current to pass through the patient’s heart to stop fibrillation – that is, to defibrillate the heart. Ironically, electric current through the heart can also cause fibrillation, depending on the amount of current, it may even cause electric shock. Currents as low as 20 mA may cause difficulty in breathing, and at 75 mA breathing may stop completely. Currents between 100 and 200 mA results in ventricular fibrillation of the heart, which means an uncoordinated and uncontrolled twitching of the heart muscles. The resulting loss in pumping action  is fatal. The defibrillator used in medical emergencies apply a large momentary voltage to the body to stop the heart and facilitate the restoration of the normal heart rhythm.

    

LAW OF CONSERVATION OF ENERGY.  

       Energy can never be created or destroyed, it maybe transformed from one form to another, but the total amount of energy never changes ( remains constant ). 

       Total energy is the sum of all forms of energy in a system : kinetic, heat, potential chemical, etc. Experiments have shown that the total energy in a closed system is always conserved.  Energy can be transferred from one system to another if one system does work on the other.  The conservation of energy can be written in the form :

        E_k + E_p + E_o = constant  ==> E_ki  +  E_pi  + E_oi  =  E_kf  +  E_pf  +  E_of

where the subscript i  and f  denote initial and final energies. E_k represents kinetic energy, E_p represents potential energy and  E_o represents all other forms of energy.  In the treatment of the equation if  the E_o is constant, then E_oi = E_of and the equation can be reduced to E_ki  +  E_pi =  E_kf  +  E_pf. The law of conservation of energy principle is very useful in solving problems. It can be applied to any closed system, where only the initial and final conditions need to be considered.

POWER AND EFFICIENCY

      Power is the time rate of doing work. It is the rate at which energy is used or expended, since work done results in energy being transferred from one system to another. The SI unit for power is the joule per second (J/s), which is called the watt in honor of James Watt ( 1736 – 1819 ).   1 J / s  = 1 watt.   Watt is a familiar unit. All light bulbs and other electric devices are rated in watts. 

      The horsepower was defined by Watt as a unit for power. He was interested in describing the rate at which his steam engine could do work and defined his unit in terms of the common source of power, the horse. He found out that on the average, the horse was doing about 550 ft-lbs of work per second. He called this unit one horsepower and measured the rate at which his steam engine could work and rated them in horsepower. 

                                                            P = W / t 

Units of Power

     In the U.S. the watt and the kilowatt are used exclusively in connection with the electric power and horsepower is reserved for  mechanical power. This  practice  is purely  a  convention  and  by  no

means necessary. The unit watt is in honor of James watt.  1 J / s  = 1 watt.   
1 kilowatt = 1000 watts 

 1 horsepower (hp) = 550 ft-lbs / sec = 33,000 ft-lbs/min = 746 watts = 0.746 kw

     In the British Engineering system, power is expressed in ft– lb  per  second but more often is given in horsepower. The origin of the Hp started when James Watt was trying to sell steam engines for the British Coal Mines. He was asked how many horses would be replaced by his engine. Watt found out that on the average, horses were doing about 550 ft-l b/s work, then he called this unit one hp. He measured the rate at which his steam engines could work and rated them in horsepower.     

     1. A 75 kg man climbs a flight of stairs 5.25 m high in 15 seconds. Determine the power developed in watts and horsepower. What is the power developed if the man is running up the stairs in  5  seconds ?

       Note :  It is no wonder that running upstairs is so stressful and causes the body to utilize its available energy very quickly. People with heart problems are warned that climbing stairs is one of the most stressful acts that they can perform.

Power Billing

     The equation P = W/ t  can be solve for work:  W = P t . The kilowatt– hour  unit used by electric companies in billing is a unit for work, where power is in kilowatt and time is in hours.  The bill is for the amount of work that has been done to the consumer. The cost of the work done is obtained by multiplying the total energy consumption by the rate per kilowatt-hour. 

                        W = P t ,          C = W r 

             where : P = power in kilowatt,  t = time in hours ,  C = cost or amount of electric bill in P 

                           r = energy rate per kilowatt-hour 

   Ex.  Determine the total energy consumed in kilowatt-hour for loads of  a) one 1.5 hp air conditioning machine operating at 6 hours per day and 30 days per month, b) two 200 watts desktop computer operating 4 hours a day and 30  days per month, c) five 20 w lamps operating 4 hours a day and 30 days per month and d) one 125 watts tv set operating 7 hours a day, 30 days per month. If energy rate is P 6.75  per kilowatt-hour, what is the bill at the end of the month? 

Force and Newton's Laws of Motion

FORCE, THE CAUSE OF ACCELERATION; NEWTON’S LAWS OF MOTION

        In 1642, several months after Galileo died, Isaac Newton was born. At age 23, Newton developed his famous laws of motion, which completed the overthrow of the Aristotelian idea that had dominated the thinking of the best minds for 2,000 years.

         Every acceleration ( change in velocity ) is caused by forces acting on a body. Conversely, if a body does not accelerate, then the total force acting on it is zero even if several forces are present. The apparently simple idea of cause and effect, that forces cause acceleration, didn’t come easily. It was and still is tempting to think of common phenomena as having no cause and simple being “the nature of things”. For example, “Why does water flow downhill?” seems stupid. Yet such question have a serious answers; in this case, the force of gravity causes water to flow downhill. The genius of Newton and others was not only in providing answers to basic questions. But also in simply being curious enough to ask basic questions.

          Force is defined intuitively as a push or a pull. If an applied force is the  only one thing acting on a body, then the body will accelerate in the same direction as the force. The strength of the force determines the magnitude of the acceleration. If several forces act on a body, then its acceleration is in the same direction as the total force and has magnitude proportional to the total force.

NEWTON’S LAWS OF MOTION

Galileo had a major influence in the study of motion. What Newton did was to write down the relationships between the force and motion in a form that could be used to predict and describe motion. Those relationships were found to apply in every circumstance where an experiment could be performed to test them and came to be known as Newton’s laws of motion. 

      The First Law : Inertia (mass). A body rest remain at rest or in motion in a straight line with a constant velocity unless acted upon by an outside force. The property of a body that causes it to remain at rest or to maintain a constant velocity is called inertia. The law was a refinement of Galileo’s idea --- in the absence of force, a moving object will continue moving. Galileo considered the tendency of things to resist change in motion as inertia. Inertia is a measure of how difficult is it to set a body into motion, or if it is already moving, how difficult is it to stop.

 

      The Second Law: The acceleration produced by forces acting on a body is directly proportional to and in the same direction as the net external force and inversely proportional to the mass of the body.

         a = F_{net ext} / m  ==>   F_{net ext} = ma ,       m = mass  and  a = acceleration

Newton’s Second law gives a precise definition of force that is consistent with our intuitive notions of a force as a push or a pull. A large force produces a large acceleration, a large mass requires a large force to make it accelerate at the same rate as a small mass, and a body will accelerate in the same direction as the net force on it.

      

      The Third Law : Action – Reaction. Whenever one body exerts a force on a second body, the second body exerts a force back on the first that is equal in magnitude and opposite in direction.  This is paraphrase as,  “For every action there is equal and opposite reaction”.  

       One force is called the action force and the other is the reaction force. In every interaction, the forces always occur in pairs. The action and the reaction pair of forces make up the interaction between two things.  We know that forces can cancel when they are equal and act in the opposite direction on the same object. Even though action and reaction are equal and oppositely directed, they do not cancel each other for  they are acting on different bodies.

       An example is a swimmer that exerts a force on the side of the pool. By Newton’s third law, the side of the pool exerts a force back on the swimmer – an external force. If friction is negligible between the swimmer and the water, she will then move in a direction opposite to the force she exerted on the side of the pool with an acceleration proportional to the force she exerted.

       Cars accelerate forward by exerting backward forces on the ground. The reaction force of the ground acts as an external force on the car in the forward direction. 

UNITS OF FORCE

 1.  Newton – is the force required to give a mass of 1 kilogram an acceleration of  1 m/ s². 

             1 newton ( N ) = 1 kg-m/ s²   

 2.  Dyne – is the force required to give a mass of 1 gram an acceleration of  1 cm/ s². 

             1 dyne = 1 g-cm/ s²  

 3.  Pound – is the force required to give a mass of 1 slug an acceleration of 1 ft/ s².  

             1 lb = 1 slug-ft/ s² = 4.448 N

 WEIGHT, FRICTION,  TENSION, AND OTHER CLASSES OF FORCES

       The weight of an object is the gravitational force exerted on it by the earth. When an object is dropped near the earth’s surface, it is accelerated by the gravitational force with an acceleration g, thus by Newton’s second law, the weight w becomes 

                                                            w = mg.       ==>    m = w/ g

       We see in this equation the relation between mass and weight : Weight is a force proportional to the mass of a body and g is the constant of proportionality. Here, g is taken to be positive, since the direction of forces are indicated with plus or minus sign. Weight depends on the location of the object, since the acceleration of gravity varies with location. As you go higher, g decreases so that weight also is decreasing. On the moon g =  1/6 ( Earth's g )

 

       Center of gravity. The force of gravity on solids can be considered to act on a single point, called center of gravity (c.g.).For symmetrical objects, c.g. is at its geometric center. For asymmetrical objects, the c.g. is closer to the more massive part of the body. A closer related concept is the center of mass (c.m.), is the point at which all of the mass in a body can be considered to be located.

      Newton’s Universal Law of Gravitation. The law states that there a force of attraction between any two masses that is proportional to the product of the masses and inversely proportional to the square of the  distance between their centers of mass.

                                                                F = G m₁ m₂ / r² 

             where  G = Newton’s Universal constant of gravitation  

                         G = 6.67 x 10^–11  N . m²/ kg²  ,  m₁  &   m₂  =  masses in kg  and

                           r = distance between the centers of mass in meter.

FRICTION

       Friction is any force that opposes every effort to start to slide or roll one body over another body. Frictional forces are specially important to us in our daily lives, for without them we could not walk or hold things with our hands; cars wouldn’t be able to start or stop; nails and screws would be useless. Frictional forces are not fundamental forces like gravity or electromagnetism, but arise as reaction to other applied forces. Friction is proportional to the force exerted by one substance on another perpendicular to the surface between them---that is, the normal force (perpendicular force). The mathematical expressions are :

          1.  f = ukF_N     ,          uk is coefficient of kinetic friction  ,   F_{N is} Normal force

          2.  f = us F_N   ,        us is coefficient of static friction

 

     Equation 1 is used for the friction between moving substance and equation 2 for stationary substances. 

 Coefficient of friction is the ratio of the force of friction f to the normal force, F_N.                                                                                           

PRINCIPLES OF FRICTION

     1. The force of friction  always act in a direction opposite to the direction of motion, for objects in relative notion --- that is, sliding or rolling.

     2. The frictional force is proportional to the normal (perpendicular ) force between the two surfaces in contact.

     3. Frictional force is approximately independent of the area of contact between the surfaces.

     4. The frictional force depends on the particular material that make up the surfaces.  

* Synovial fluid – a fluid which looks like blood plasma which lubricates the joints and limbs of the body.

 ADVANTAGES OF FRICTION

     1.  Walking would be impossible without friction.

     2.  Pulley driven machines depend on friction for their operation.

     3.  Friction prevents belts from slipping off their pulley.

     4.  Friction between the tires and the road prevents skidding of vehicles.

     5.  Clutch, bolts and nuts, nails, screws, matches, brakes, etc. depends on friction.

 DISADVANTAGES OF FRICTION

    1.  Wearing out of parts of machines, thus causing extra expenses for maintenance.

   2.  It causes expansion on machine parts and heat losses due to friction thus reducing the efficiency of machines.

TENSION

      A tension is any force carried by a flexible string, rope, cable, chain, etc. Because the medium carrying the force is flexible, it can only pull and can exert no force except along its length. Tension comes from a Latin word meaning “to stretch thin”. In muscle systems the fibrous cords that carry forces exerted by muscles to other parts of the body are called tendons. Tension is due to the cohesive atomic and molecular electromagnetic forces acting in a string.

      For a body suspended on a string with zero or constant speed upward or downward, the tension is given by  T = w = mg.  If the body accelerates downward on a string, the tension is given by  T + ma  = mg  and if the body has an upward acceleration on a string, the tension is given by  T =  mg +  ma.            

                          

  Problems :     

1. Determine the weight of a 70 kg person on earth. On the moon, g is 1/6 of the earth’s g, what is the weight of this person ?

2. Determine the mass of a box if a force of 80 N is able to accelerate it at 1.25 m/ s².   

3. Calculate the mass of a flea in grams if its weight is 5 x 10^{– 6} N.

4. Find the acceleration of a rocket with mass of 1.2 x 10 ⁶  kg  if its engine exerts a net force of  2 x 10 ⁶ N.

5. A 70 kg gymnast climbs on a rope. Determine the tension in the rope if (a) he climbs at constant  speed,  (b) he has an upward acceleration of 0.5 m/ s² ; and  (c)  he goes downward with a downward acceleration of 0.5 m/ s².

6. Determine the force of gravitation between the earth and the sun and between the earth and the moon. 
   m_E  = 5.99 x 10²⁴ kg ,    m_S =  1.99 x 10 ³⁰  kg  ,     m_M = 7.36 x 10 ²² kg

         S earth–sun = 149.6 x 10⁹ m ,  S  earth–moon = 3.84 x 10⁸ m,   
         Radius of  the earth = 6.367 x 10⁶ m 

    7.  A man weighs a fish of mass m on a spring scale attached to the ceiling of a elevator.  Show that if the elevator accelerates in either direction , the spring scale gives a reading different from the weight of  the fish.  What is the reading on the scale if the elevator moves up or down at constant speed?