Work Power and Energy

Experiment 7: Work, Power and Energy Pocholo Luis P. Mendiola Department of Mathematics and Physics College of Science, University of Santo Tomas España, Manila Philippines Abstract The experiment employed basic principles of work, power and energy which include the change of both the kinetic and potential energy of a body under the influence of gravity, the conservation of mechanical energy, and the work of a body running against the time paramenter, also known as the power P. The conservation of mechanical energy was verified through a graphical anaylsis of the object under free fall. In addition, simple demonstration of the power output by the use of staircase was also employed in the experiment. 1. Introduction Nowadays, energy conservation is undoubtly the single most important idea in physics. Strangely enough, although the basic idea of energy conservation was familiar to scientists from the time of Newton onwards, this crucial concept only moved to the centre-stage of physics in about 1850 when scientists realized that heat was a form of energy [4]. Energy can take many different forms: potential, kinetic, thermal, chemical, electrical, nuclear, etc. In fact, everything that we observe aroud us represents one of the numerous manifestations of energy. However, all these processes leave the total amount of energy in the Universe invariant; that is, the total energy of a system is

unchanged regardless of any transformation or even rotation of the coordinate systems. In other words, whenever, and however, energy is transformed from one form into another, it is always conserved. For a closed system; (i.e., a system which does not exchange energy with the rest of the Universe,) implies that the total energy of the system in question must remain constant in time. Work, on one hand, as used in physics has a narrower meaning than it does in everyday life. First, it only refers to physical work, of course, and second, something has to be accomplished. Technically, work is done when a force pushes something and the object moves some distance in the direction it’s being pushed or pulled. The quantitative idea of how much work is done is usually expressed in units to measure it. In addition, the rate at of work done also known as the power, is commonly used particularly in electricity bill which usually expressed in terms of kilowatt-hours or kWh. The objectives of the experiment are to demonstrate the conservation of mechanical energy, to measure the change in kinetic and potential energies as ball moves in free fall, and to determine the power output when going up and down the staircase.

2. Theory Work In physics, work is defined as the dot (scalar) product of force F and displacement s, that is, [eq. 2.0] where is the angle between the applied force F and the displacement s The force and displacement vectors are multiplied together in such a way that the product yields a scalar. Thus, work is not a vector, and has no direction associated with it. Since work is the product of force and displacement, it has units of newtonmeters, or joules (J). A joule is the work done by applying force on one newton through a displacement of one meter [2]. Mathematically, no work is done if the force F and the displacement s are perpendicular to each other. This is because the angle between these two vectors is 90o. Since cos(90o) = 0, the resulting value for work would be 0. Work is a measure of effort expended by a force moving an object from one point to another point [a]. If the force varies then the total amount of work done is determined by a definite integral ∫

( )

[eq. 2.1] It is also interesting to note that work is not a measure of how tired you are after perform the work. It is a measure of the

product of the force that was applied in the direction of the displacement. Work is also a measure of the energy that was transferred while the force was being applied. Power Work can be done slowly or quickly, but the time taken to perform the work doesn’t affect the amount of work which is done since there is no element of time associated with it. However, if you do the work quickly, you are operating at a higher power even that if you do the work slowly. Power is defined as the rate at which the work is done or the rate of energy transfer. The equation for power is

[eq. 2.2] where P = Power, velocity.

= work done, v =

When a quantity of work is done during a time interval , the average work done per unit time or average power Pav is defined to be [eq. 2.4] The rate at which work is done might not be constant. Redefining, the instantaneous power P as the quotient in [eq. 2.4 ]. As approaches zero [eq. 2.5] The SI unit of power is watt (W), named for the English inventor James Watt. One watt equals 1 joule per second: 1 J = 1

J/s. The kilowatt (1 kW = 103 W) and the megawatt (1 MW = 106 W) are also commonly used. In the British system, worlo is expressed in foot-pounds, and the unit of power is the foot-pound per second [3]. A larger unit called the horsepower (hp) is also used 1 hp = 746 W = 0.746 kW The watt is a familiar unit of electrical power. The kilowatt-hour (kWh) is the usual commercial unit of electrical energy. One kilowatt-hour is the total work done in 1 hour (3600 seconds) when the power is 1 kilowatt (103 J/s). In Classical Mechanics, power can also be expressed in terms of force and velocity [3]. Suppose that a force F on a body while it undergoes a vector displacement . If the F|| is the component of F tangent to the path (parallel to ), then the work done by the force is = F|| , The average power is

Energy Energy is the ability to do work, and when work is done, there is always a transfer of energy. Energy can take on many forms, such as potential energy, kinetic energy, and heat energy. The unit for energy is the same as the unit for work, the joule (J). This is because the amount of work done on a system is exactly equal to the change in energy of the system. This is called the work-energy theorem. Potential Energy Potential energy is the energy a system has because of its position or configuration [4]. For instance, stretching a rubber band means you store energy in the rubber band as elastic potential energy. Potential energy generally has two forms. On one hand, it is the gravitational potential energy Ugrav. It is defined as the product of the body’s weight (mg) and its height (y) above the ground Ugrav = mgy

[eq. 2.6] Instantaneous power P is the limit of the above expression as :

[eq. 2.7] Where v is the magnitude of the instantaneous velocity. It can also be expressed in terms of the dot (scalar) product: [eq. 2.8]

[eq. 2.9]

This type of potential energy is associated with the body’s weight and its height above the ground. On the other hand, there is the elastic potential energy Uel. Elastic potential energy arises when there are situations in which the potential energy is not gravitational in nature. It is defined as half of the product of the force constant of the spring k and the square of its displacement x [eq. 2.10]

Mathematically, the potential energy is a path-independent or conservative physical quantity. For instance, consider a body moving in a conservative force-field f(r), Picking some point O arbitrarily in the field, we can define a function U(r) which possesses a unique value at every point in the field [4]. The value of this function is associated with some general point R is simply ( )

∫

Fig. 2.0 The derivation of the Work-Energy Theorem (Photo credit: http://homepages.wmich.edu/~kaldon/classes/ph20522-KE-Derivation-Calculus.gif)

Conservation of Mechanical Energy

[eq. 2.11] In other words, U(R) is just the energy transferred to the field (i.e., minus the work done by the field) when the body moves from point O to point R. The value of U at point O is zero: i.e., U(O) = 0. The above definition uniquely specifies U(R), since the work done when a body moves between two points in a conservative forcefield is independent of the path taken between these points [4]. Kinetic Energy The kinetic energy (KE) is an energy of an object has because it is moving. The KE of a moving object depends on its mass m and the square of its velocity v [eq. 2.12] But in order for a mass to gain KE, work must be done on the mass to push it up to a certain speed or to slow it down. The work-energy theorem states that the change in KE of an object is exactly equal to the work done on it [2], assuming there is no change in the object’s potential energy.

When work is done on a system, the energy of that system changes from one form to another, but the total amount of energy remains the same. The total energy therefore can be said that is is conserved, that is, remains constant during any process. This is also called the law of conservation of energy. For a closed system, (i.e., a system which does not exchange energy with the rest of the Universe), implies that the total energy of the system in question must remain constant in time [4]. The kinetic energy (KE) represents energy the mass possesses by virtue of its motion. Likewise, potential energy (U) represents energy the mass possesses by virtue of its position. The total energy E of the system therefore can be written ME = KE + U = constant

[eq. 2.13]

In other words, the increase in the KE of the body, as it moves from some point to another point, is equal to the decrease in the U evaluated between these same two points.

It is clear that E is a conserved quantity. Although the KE and U of the object varies, its total energy remains the same. The principle of conservation of mechanical energy also states that energy of an isolated system can be neither created nor destroyed but can be transformed to other forms of energy [2]. In other words, the following must hold: MEbefore = MEafter

[eq. 2.14]

3. Methodology In the experiment, two activities were performed. One of which is the power output which first requires the determination of work by each member when going up and down the staircase. Secondly, the demonstration of the conservation of mechanical energy of the tossed ball by the use of the motion detector computer software Vernier Logger Pro(c).

After obtaining all the relevant data, the power output P of each member was calculated by multiplying the force F and the displacement s (in this case, htotal = s) and dividing it overall by the time taken to go up and down the staircase. In addition, the most “powerful” member of the group was determined by the largest numerical value of the calculated power output. Activity 2: Energy a Tossed Ball The graphical curves were predicted and sketched of potetential energy versus time, kinetic energy versus time, and the combination of the two – the total mechanical energy versus time of a ball thrown vertically upward from a height of 50.0 cm. In able to compare the predicted graphs, the motion detector computer software Vernier Logger Pro(c) with the file “16 Energy of a Tossed Ball” was used. The ball was tossed straight upwards while holding it 50.0 cm above the motion detector until it began to collect data.

Activity 1: Power 4. Results and Discussion Initially, the weight of each member was determined. This will generally serves to be the force F that will be used for the calculation of the work done. Each member of the group was asked to go up and down the staircase of the Main Building and recorded the time for each. Afterwards, the vertical distance h of the staircase was determined using a meter stick by summing up the height of each of the successive risers: htotal = h1 + h2 + ... + hn

[eq. 3.0]

Activity 1 Vertical distance between second floor and third floor = 2.513 m Table 1. The work done and power output of Each Member of the group in going up and down the staircase. Member 1 2 3 Weight (N) Work in going up (J) Time to go up (s) Power output in going up(W)

588 N 1477.64 J 5.6 s

401.8 N 1009.72 J 29.1 s

519.4 N 1305.25 J 5.6 s

263.86 W

34.70 W

233.08 W

Work in going down (J) Time to go down (s) Power output in going down (W)

1477.64 J 5.5 s

1009.72 J 9.9 s

1305.25 J 7.8 s

268.66 W

101.991 W

167.34 W

From Table 1, the work and power output changes per member. Ideally, the power to go down shoud be lesser than to go up because of safety issues. However, one gathered data shows that going down requires less power. This is because of the pull of gravity. The gravitational pull g naturally acts downward hence, it is easier to follow its direction than to oppose it.

c. Fig. 4.2 Graph of total mechanical energy vs. time

d. Fig. 4.3 Total mechanical energy plot by the use of Vernier Logger Pro(c)

Activity 2 Predictions: a. Fig. 4.0 Graph of potential energy vs. time

b. Fig. 4.1 Graph of kinetic energy vs. time

From the graphs provided above, the theoretical curve plots are a., b., c. With the use of the motion detector computer software Vernier Logger Pro(c), the curve from the first second of d. is somewhat similar. Although the curve seems to be messy between the time 1s and 2s. This is because there are several contributing factors that affect the entirety of the graph. For instance, the ball may not fall straigthly downward as it was tossed straightly upward.

5. Conclusion Work done is directly proportional to the applied force in the direction of the displacement. Since work done is equivalent to the product of force and displacement, we can readily conclude that as the work done is increasing given the displacement constant, the force is also increasing. In addition, since the power output is equivalent to the work done over the time taken to do that, given a constant time, it can be said that the power expended is directly proportional to the work done. Also, for the energy, it is really neither created nor destroyed. From the graphs provided, the relationship between the kinetic and potential energies are opposite; that is, it’s either lose or gain. Although they vary when the object moves, their sum is still the same (as stated in the conservation of mechanical energy) regardless of any situations (assuming it is in a closed system).

6. Applications 1. Compare the work that you do when you go upstairs to the work you do in going downstairs. Based on this, can you explain why it is more difficult to go upstairs than downstairs? Usually, the work done is the same because the weight is invariant in a particular location. Furthermore, it is more difficult to go upstairs because you are opposing the natural direction of gravity which is acted downwards (g = 9.8 m/s2).

2. A certain professor finds it easy to go upstairs from the ground floor to the third floor of the main Building by going up the second floor using the main stairs, walking along the corridor of the accounting division and using the side stairs to go to the third floor. Is there a basis to this from the point of view of physics? Yes, and it is because of the steepiness of the stairs. It is much difficult to use a stair with more inclination as you tend to be more careful to avoid any accident/s which results to a greater effort and longer time.

3. It is 5 minutes before your 7:00 am class in the fourth floor and you are still in the ground floor. Will you run or walk upstairs in order not to be late? Assume that your power output is 15 watts and 20 watts when walking and running, respectively. The vertical distance between the ground floor and the fourth floor is 12 m and that you weigh 750N.

I would rather run because the resulting time will only be 450s or about 7.5 minutes only than walking which will be 600s or about 10 min.

4. An object is thrown vertically up. Neglecting air resistance, how is the change in the potential energy of the

object related the change in its kinetic energy? As the object is thrown up, it is slowing down. As a result, the kinetic energy decreases. Moreover, when the object slows down due to the force of gravity pushing down on it. Hence, the gravitational potential energy increases. For upward to downward, it is lose of KE – gain of PEgrav. 7. References [1] Stewart, J. (2003). Early Transcendentals Single Variable Calculus. Canada, CA: Thomson Learning. [2] Zeitlin, J. (2003). SAT II: Physics. Canada, CA: Kaplan Publishing. [3] Ford, A., Freedman, R., Young, H. (2012). Sears and Zemansky’s University Physics with Modern Physics. New York, NY: Pearson Learning. [4] Fitzpatrick, R. (n.d.). Classical Mechanics. Retrieved online from farside.ph.utexas.edu/teaching.../301.pdf