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LAB - WORK AND ENERGY
Energy is the only life and is from the Body; and Reason is the bound or outward circumference of energy. Energy is eternal delight. William Blake
OBJECTIVES To extend the intuitive notion of work as physical effort to a formal mathematical definition of work, W, as a function of both the force on an object and its displacement. To develop an understanding of how the work done on an object by a force can be measured. To understand the concept of power as the rate at which work is done. To understand the concept of kinetic energy and its relationship to the net work done on an object as embodied in the workenergy principle. To understand the concept of potential energy. To understand the concept of mechanical energy of a system. To investigate situations where mechanical energy is conserved and those where it is not. OVERVIEW In your study of momentum in the previous lab you saw that while momentum is always conserved in collisions, apparently different outcomes are possible. For example, if two identical carts moving at the same speed collide head-on and stick together, they both end up at rest immediately after the collision. If they bounce off each other instead, not only do both carts move apart at the same speed but in some cases they can move at the same speed they had coming into the collision. A third possibility is that the two carts can explode as a result of springs being released (or explosives!) and move faster after the interaction than before. Two new concepts are useful in further studying various types of physical interactions work and energy. In this lab, you will begin the process of understanding the scientific definitions of work and energy, which in some cases are different from the way these words are used in everyday language. We will introduce the principal of Conservation of Energy.
University of Virginia Physics Department PHYS 142W, Spring 2009 Modified from P. Laws, D. Sokoloff, R. Thornton
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You will begin by comparing your intuitive, everyday understanding of work with its formal mathematical definition. You will first consider the work done on a small point-like object by a constant force. There are, however, many cases where the force is not constant. For example, the force exerted by a spring increases the more you stretch the spring. In this lab you will learn how to measure and calculate the work done by any force that acts on a moving object (even a force that changes with time). Often it is useful to know both the total amount of work that is done, and also the rate at which it is done. The rate at which work is done is known as the power. Energy (and the concept of conservation of energy) is a powerful and useful concept in all the sciences. It is one of the more challenging concepts to understand. You will begin the study of energy in this lab by considering kinetic energya type of energy that depends on the velocity of an object and to its mass. By comparing the change of an object`s kinetic energy to the net work done on it, it is possible to understand the relationship between these two quantities in idealized situations. This relationship is known as the work--energy principle. You will study a cart being pulled by the force applied by a spring. How much net work is done on the cart? What is the kinetic energy change of the cart? How is the change in kinetic energy related to the net work done on the cart by the spring? Suppose you lift an object steadily at a slow constant velocity near the surface of the Earth so that you can ignore any change in kinetic energy. You must do work (apply a force over a distance) to lift the object because you are pulling it away from the Earth. The lifted object now has the potential to fall back to its original height, gaining kinetic energy as it falls. Thus, if you let the object go, it will gain kinetic energy as it falls toward the Earth. It is very useful to define the gravitational potential energy of an object at height y (relative to a height y 0 0 ) as the amount of work needed to move the object away from the Earth at constant velocity through a distance y . If we use this definition, the potential energy of an object is a maximum when it is at its highest point. If we let it fall, the potential energy becomes smaller and smaller as it falls toward the Earth while the kinetic energy increases as it falls. We define the mechanical energy as the sum of these two energies. We can now think of kinetic and potential energy to be two different forms of mechanical energy. Is the mechanical energy constant during the time the mass falls toward the Earth? If it is, then the amount of mechanical energy doesn`t change, and we say that mechanical energy is conserved. If mechanical energy is conserved in other situations, we might be able to hypothesize a law of conservation of mechanical energy as follows: In certain situations, the sum of the kinetic and potential energy, called the mechanical energy, is a constant at all times. It is conserved. The concept of mechanical energy conservation raises a number of questions. Does it hold quantitatively for falling masses? Is the sum of the calculated potential and kinetic energies exactly the same number as the mass falls? Can we apply a similar concept to masses experiencing other forces, such as those exerted by springs? Perhaps we can find another definition for elastic potential energy for a mass--spring system. In that case
University of Virginia Physics Department PHYS 142W, Spring 2009 Modified from P. Laws, D. Sokoloff, R. Thornton
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could we say that mechanical energy will also be conserved for an object attached to a spring? Often there are frictional forces involved with motion. Will mechanical energy be conserved for objects experiencing frictional forces, like those encountered in sliding? You will explore the common definition of gravitational potential energy to see if it makes sense. You will then measure the mechanical energy, defined as the sum of gravitational potential energy and kinetic energy, to see if it is conserved when the gravitational force is the only force acting. Next, you will explore a system where the only net force is exerted by a spring and see the definition of elastic potential energy. You will measure the mechanical energy of this system and see if it is conserved. Finally, you will explore what effects sliding frictional forces or air resistance forces have on systems. You will explore whether or not mechanical energy is still conserved in such systems. INVESTIGATION 1: THE CONCEPTS OF PHYSICAL WORK AND POWER While you all have an everyday understanding of the word work as being related to expending effort, the actual physical definition is very precise, and there are situations where this precise scientific definition does not agree with the everyday use of the word. You will begin by looking at how to calculate the work done by constant forces, and then move on to consider forces that change with time. Let`s begin with a prediction that considers choosing among potential real-life jobs. Prediction 1-1: Suppose you are president of the Load n` Go Company. A local college has three jobs it needs to have done and it will allow your company to choose one before offering the other two jobs to rival companies. All three jobs pay the same total amount of money.
Which one would you choose for your crew? Explain why.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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The following activities should help you to see whether your choice makes the most sense. You will need the following: 5 N spring scale 2 m motion track that can be inclined track inclinometer digital mass scale motion cart with no friction pad two kg masses clamps and rods In physics, work is not simply effort. In fact, the physicist`s definition of work is precise and mathematical. To have a full understanding of how work is defined in physics, we need to consider its definition for a very simple situation and then enrich it later to include more realistic situations. If a rigid object or point mass experiences a constant force along the same line as its motion, the work done by that force is defined as the product of the force and the displacement of the center of mass of the object. Thus, in this simple situation where the force and displacement lie along the same line
W Fx x
where W represents the work done by the force, Fx is the force, and x is the displacement of the center of mass of the object along the x axis. Note that if the force and displacement (direction of motion) are in the same direction (i.e., both positive or both negative), the work done by the force is positive. On the other hand, a force acting in a direction opposite to displacement does negative work. For example, an opposing force that is acting to slow down a moving object is doing negative work. Question 1-1: Does effort necessarily result in physical work? Suppose two people are in an evenly matched tug of war. They are obviously expending effort to pull on the rope, but according to the definition are they doing any physical work as defined above? Explain.
Activity 1-1: Work When the Force and Displacement Lie Along the Same Line and When They Don't In this activity you will measure the force needed to pull a cart up an inclined ramp using a spring scale. You will examine two situations. First, you will exert a force parallel to the surface of the ramp, and then you will exert a force at an angle to the ramp. You will then be able to see how to calculate the work when the force and displacement are not in the same direction in such a way that the result makes physical sense.
University of Virginia Physics Department PHYS 142W, Spring 2009
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Lab 8 - Work and Energy
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1. Set up the cart and ramp as shown in the diagram below. Add two kg masses to the cart. Weigh mass of cart and the additional masses. Attach the hook on the spring scale to the screw on top of the cart. Support one end of the ramp so that it is inclined to an angle of about 10.
Mass of cart: ______________ g
mass #1: ______________ g
mass #2: ______________ g
Total mass of cart & additional masses: ______________ kg
45
10
2. Find the force needed to pull the cart up the ramp at a constant velocity. Pull the cart so that the spring scale is always parallel to the ramp. Pull the cart along the ramp and write down the average force on the spring scale.
Average force pulling parallel to ramp: ______________ N
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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Lab 8 - Work and Energy
Prediction 1-2: Suppose that the force is not exerted along the line of motion but is in some other direction, like at an angle of 45 to the ramp. If you try to pull the cart up along the same ramp in the same way as before (again with a constant velocity), only this time with a force that is not parallel to the surface of the ramp, will the force probe measure the same force, a larger force, or a smaller force?
3. Now test your prediction by measuring the force needed to pull the cart up along the ramp at a constant velocity, pulling at an angle of about 45 to the surface of the ramp. Measure the 45 angle with a protractor. Measure the force on the spring scale as you pull the cart up at a slow constant speed as shown in the diagram above. Be sure the cart does not lift off the surface of the ramp.
Average force pulling at 45 to the surface: ___________N Question 1-2: Discuss the difference between the average force (measured by the spring scale) when the cart was pulled at 45 to the surface and the average force when the cart was pulled parallel to the surface.
It is the force component parallel to the displacement that is included in the calculation of work. Thus, when the force and displacement are not parallel, the work is calculated by
W Fx x ( F cos ) x F x
Question 1-3: Discuss how well your observations support this cosine dependence as a reasonable way to calculate the work.
Sometimes more than just the total physical work done is of interest. Often what is more important is the rate at which physical work is done. Average power, P , is defined as
University of Virginia Physics Department PHYS 142W, Spring 2009 Modified from P. Laws, D. Sokoloff, R. Thornton
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the ratio of the amount of work done, so that
W
, to the time interval,
W t
t
, in which it is done,
P
If work is measured in joules and time in seconds, then the fundamental unit of power is the joule/second, and one joule/second is defined as one watt. A more traditional unit of power is the horsepower, which originally represented the rate at which a typical work horse could do physical work. It turns out that 1 horsepower (or hp) = 746 watts Those of you who are car buffs know that horsepower is used to rate engines. The engine in a high-performance car can produce hundreds of horsepower. INVESTIGATION 2: WORK DONE BY CONSTANT AND NON-CONSTANT FORCES Few forces in nature are constant. A good example is the force exerted by a spring as you stretch it. In this investigation you will see how to calculate work and power when a non-constant force acts on an object. You will start by looking at a somewhat different way of calculating the work done by a constant force by using the area under a graph of force vs. position. It turns out that, unlike the equations we have written down so far, which are only valid for constant forces, the method of finding the area under the graph will work for both constant and changing forces. The additional equipment you will need includes the following: motion detector spring rod support for force probe 200 g mass motion cart with no friction pad index card, 4 x 6 masking tape Activity 2-1: Work Done by a Constant Lifting Force In this activity you will measure the work done when you lift an object from the floor through a measured distance. You will use the force probe to measure the force and the motion detector to measure distance.
University of Virginia Physics Department PHYS 142W, Spring 2009
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> 20 cm
1. The motion detector should be on the floor, pointing upward. Use the broad beam setting on the motion detector. 2. Open the experiment file called L08.2-1 Work in Lifting. This will allow you to display velocity and force for 5 s. 3. Use masking tape to tape an index card on the bottom of a 200 g mass. This will enable the motion detector to more easily see the position of the mass. 4. Zero the force probe with the hook pointing vertically downward. Then hang the 200 g mass from its end. The index card must be relatively level or you will receive spurious results. Reattach the card if it is not level. Begin graphing and then lift the force probe by hand with the mass attached at a slow, constant speed through a distance of about 1.0 m starting at least 20 cm above the motion detector. 5. Keep trying until you have a set of graphs in which the mass was moving at a reasonably constant speed and had no spurious distance measurements. 6. Print one set of graphs for your group report. Question 2-1: Did the force needed to move the mass depend on how high it was off the floor, or was it reasonably constant?
7. You should find a force vs. position graph minimized. Click on Force vs P graph and bring it up on the screen. 8. Print out one graph for your group report.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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9. Use the statistics features of the software to find the average force over the distance the mass was lifted. Record this force and distance below. Use the Smart Tool to find the corresponding distance.
Average force: _____________ N
Distance lifted: _____________ m
10. Calculate the work done in lifting the mass. Show your calculation.
Work done:____________ J 11. Notice that force times distance is also the area of the rectangle under the force vs. position graph. Find the area under the curve by using the area routine under appropriate lines.
Area under force vs. position graph:___________ J Question 2-2: Discuss how well the two calculations of the work agree with each other.
Comment: This activity has dealt with the constant force required to lift an object against the gravitational force at a constant speed. The area under the force vs. position curve always gives the correct value for work, even when the force is not constant. Activity 2-2: Work Done by a Non-constant Spring Force In this activity you will measure the work done when you stretch a spring through a measured distance. First you will collect data for force applied by a stretched spring vs. distance the spring is stretched, and you will plot a graph of force vs. distance. Then, as in Activity 2-1, you will be able to calculate the work done by finding the area under this graph. Comment: We assume that the force measured by the force probe is the same as the force applied by the cart to the end of the spring. This is a consequence of Newton's third law. We have set the force probe to indicate the force of our hand. 1. Set up the ramp, cart, motion detector, force probe, and spring as shown in the diagram. Pay careful attention to your Instructor who will tell you how to mount the springs so they will not be bent. Use the narrow beam on the motion detector.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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Lab 8 - Work and Energy
> 20 cm
2. Be sure that the motion detector sees the cart over the whole distance of interestfrom the position where the spring is just un-stretched to the position where it is stretched about 1.0 m. 3. Open the experiment file called L08.2-2 Stretching Spring. 4. Zero the force probe with the spring hanging loosely. Then begin graphing force vs. position as the cart is moved by hand slowly towards from the motion detector until the spring is stretched about 1.0 m. [Keep your hand out of the way of the motion detector.] 5. Print out one set of graphs for your group report. Question 2-3: Compare this forceposition graph to the one you got lifting the mass in Activity 2-1. Is the spring force a constant force? Describe any changes in the force as the spring is stretched.
Question 2-4: Describe how you can use the equation W Fx x for the calculating work done by a non-constant force like that produced by a spring?
6. Use the area routine in the software to find the work done in stretching the spring.
Area under force vs. position graph: ___________ J INVESTIGATION 3: KINETIC ENERGY AND THE WORK--ENERGY PRINCIPLE We will begin with an exploration of the definition of kinetic energy. Later, we will return to this method of measuring the area under the force vs. position graph to find the work, and we will compare the work done to changes in the kinetic energy.
University of Virginia Physics Department PHYS 142W, Spring 2009 Modified from P. Laws, D. Sokoloff, R. Thornton
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What happens when you apply an external force to an object that is free to move and has no frictional forces on it? According to Newton's second law, it should experience an acceleration and end up moving with a different velocity. Can we relate the change in velocity of the object to the amount of work that is done on it? Consider a fairly simple situation. Suppose an object is lifted through a distance and then allowed to fall near the surface of the Earth. During the time it is falling, it will experience a constant force as a result of the attraction between the object and the Earth glibly called gravity or the force of gravity. You discovered how to find the work done by this force in Investigations 1 and 2. It is useful to define a new quantity called kinetic energy. You will see that as the object falls, its kinetic energy increases as a result of the work done by the gravitational force and that, in fact, it increases by an amount exactly equal to the work done. Comment: When an object moves, it possesses a form of energy because of the work that was done to start it moving. This energy is called kinetic energy. You should have discovered that the amount of kinetic energy increases with both mass and speed. The kinetic energy is defined as being proportional to the mass and the square of the speed. The mathematical formula is
KE
1 2
mv
2
The unit of kinetic energy is the joule (J), the same as the unit of work. When you apply a net force to an object, the object always accelerates. The force does work and the kinetic energy of the object changes. The relationship between the work done on the object and the change in its kinetic energy is called the work-energy principle. In short, the work-energy principal states that the net work (considering all forces acting on the object) is equal to the change in the object`s kinetic energy:
W net KE
In the next activity, you will examine the work-energy principle by doing work on a cart with a spring and comparing this work to the change in the cart`s kinetic energy. Activity 3-1: Work--Energy Principle In addition to the material before, you will also need: two 0.5 kg bar masses to put on cart 1. Set up the ramp, cart, motion detector, force probe, and spring as shown in the diagram that follows. Place the two 0.5 kg bar masses on the cart.
> 20 cm
2. Open the experiment file called L08.3-1 Work--Energy.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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Lab 8 - Work and Energy
3. Verify that the motion detector sees the cart over the whole distance of interestfrom the position where the spring is stretched about 1.0 m to the position where it is just about un-stretched. 4. Measure the mass of the cart and bar masses (separately) and enter the sum in the formula for kinetic energy. Click on the calculator screen to do so. [The default value for the mass is 1.477 kg.] 5. Click on Accept twice (once for to accept the value and once to accept the new formula) if you change the mass value. Mass of cart and bar masses: _____________ kg 6. Zero the force probe with the spring hanging loosely. Then pull the cart along the track so that the spring is stretched about 1.0 m from the un-stretched position. 7. Begin graphing, and release the cart, allowing the spring to pull it back at least to the un-stretched position. Catch the cart before it crashes into the force probe! 8. When you get a good set of graphs, print out one set of graphs for your group report. Note that the top graph displays the force applied by the spring on the cart vs. position. It is possible to find the work done by the spring force for the displacement of the cart between any two positions. This can be done by finding the area under the curve using the area routine in the software, as in Activities 2-1 and 2-2. The kinetic energy of the cart can be found directly from the bottom graph for any position of the cart. 9. Find the kinetic energy of the cart after it is released from the initial position (equal to the change in kinetic energy as the initial kinetic energy was zero) to several (at least three) different final positions. Use the analysis feature of the software. Also find the work done by the spring up to that position. 10. Record these values of work and change in kinetic energy in Table 3-1. Table 3-1 Kinetic energy (J) Work done (J)
Question 3-1: Discuss how well your results confirm the work-energy principle.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 8 - Work and Energy
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INVESTIGATION 4: GRAVITATIONAL POTENTIAL ENERGY Suppose that an object of mass m is lifted slowly through a distance y. To cause the object to move upward at a constant velocity, you will need to apply a constant force upward just equal to the gravitational force, which is downward. We choose to define the gravitational potential energy, G P E of an object of mass m to be equal to the work done against the gravitational force to lift it:
GPE m gy
A system in which the gravitational force is essentially the only net force is a cart with very small friction moving on an inclined ramp. You can easily investigate the mechanical energy for this system as the cart rolls down the ramp. In addition to the equipment you have been using, you will need the following: motion cart track inclinometer clamps and rods 2-m motion track friction pad Activity 4-1: Gravitational Potential, Kinetic, and Mechanical Energy of a Cart Moving on an Inclined Ramp 1. Set up the ramp and motion detector as shown below. Use the narrow beam on the motion detector. The ramp should be inclined at an angle of 10 above the horizontal. Use the inclinometer to measure the angle. Ask your TA for help if you need it. Make sure that the friction pad on the cart has been removed for this activity.
s s0 motion detector H
L 10
y
y0
Remember that potential energy is relative, and we can set the zero of potential energy anywhere we want it to be. In this case, we could set it to be zero at the bottom of the ramp or the top of the ramp. Let`s simply agree to set it to be zero at the bottom. Then, when we hold the cart at rest at the top of the ramp at a height of y 0 from the table, the gravitational potential energy is m gy 0 . As it rolls down the ramp, this potential energy decreases. Its value at any position y is given by
GPE m gy
2. We can find an equation for G P E in terms of S , the position measured by the motion detector along the ramp, and L , the length of the ramp. Look carefully at the above figure. If S is the distance along the ramp away from the motion detector at time t , the equation for G P E is:
GPE mg L S sin
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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Lab 8 - Work and Energy
3. Measure the mass of the cart.
Mass: ____________ kg 4. Open the experiment file called L08.4-1 Inclined Ramp. 5. Enter the mass of the cart into the formula for kinetic energy. [Again, you must Accept twice.] Check the formulas for G P E and K E that you find on the calculator screen to make sure you agree that you are calculating the correct quantities. 6. Notice that mechanical energy is calculated as M E
GPE KE
.
Prediction 4-1: As the cart rolls down the ramp, how will the kinetic energy change? How will the gravitational potential energy change? How will the mechanical energy change?
7. Verify that the motion detector sees the cart all the way along the ramp and that the ramp is set at a 10 angle. 8. Data Studio will again utilize auto start and stop in this experiment. The range is 0.4 - 1.8 m relative to the motion detector at the top. Hold the cart at the top of the ramp about 20 cm from the motion detector and START the experiment. Release the cart and catch it at the bottom right before it slams into something. 9. Print out one set of graphs for your group report. Question 4-1: Compare your graphs to your prediction above. How are they similar and how are they different?
Comment: The mechanical energy, the sum of the kinetic energy and gravitational potential energy, is said to be conserved for an object moving only under the influence of the gravitational force. That is, the mechanical energy remains constant throughout the motion of the object. This is known as the conservation of mechanical energy.
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 8 - Work and Energy
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Activity 4-2: There and Back Again Prediction 4-2: Suppose that the cart is given a push up the ramp and released. It moves up, reverses direction, and comes back down again. How will the kinetic energy change? How will the gravitational potential energy change? How will the mechanical energy change? Describe in words.
Test your predictions: 1. Open the experiment file called L08.4-2 Inclined Ramp. Enter the cart mass as before. 2. Hold the cart at the bottom of the ramp and START the experiment. [Do not put your hand between the cart and the motion detector.] Give the cart a push up the ramp. Stop the cart when it comes down again close to the bottom. The auto start and stop will again be used to collect the data. The range is again 40 - 180 cm relative to the motion detector. 3. Print out one set of graphs for your group report. Question 4-2: How does the mechanical energy change as the cart rolls up and down the ramp? Does this agree with your prediction? Discuss.
Activity 4-3: Mechanical Energy and Friction Prediction 4-3: Suppose that there is also a frictional force acting on the cart in addition to the gravitational force. Then as the cart rolls down the ramp, how will the kinetic energy change? How will the gravitational potential energy change? How will the mechanical energy change?
University of Virginia Physics Department PHYS 142W, Spring 2009
Modified from P. Laws, D. Sokoloff, R. Thornton
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Lab 8 - Work and Energy
Test your predictions: 1. Attach the friction pad to the bottom of the cart and adjust it so that there is a reasonable amount of friction between the pad and the ramp, but so that the cart still easily rolls down the ramp when released. 2. In this experiment you will again release the cart from the top of the inclined ramp. Open the experiment file called L08.4-3 Inclined Ramp. Enter the cart mass as before. 3. Using exactly the same setup as Activity 4-2, graph K E , G P E , and mechanical energy as the c...