Objectives

• Use a Motion Detector and a force sensor to measure the position and force on a hanging mass, a spring, and a dynamics cart.
• Determine the work done on an object using a force vs. distance graph.
• Use the Motion Detector to measure velocity and calculate kinetic energy.
• Compare the work done on a cart to its change of mechanical energy.

Introduction

Work is a measure of energy transfer. In the absence of friction, when positive work is done on an object, there will be an increase in its kinetic or potential energy. In order to do work on an object, it is necessary to apply a force along or against the direction of the object’s motion. If the force is constant and parallel to the object’s path, work can be calculated using

W = F · s

where F is the constant force and s the displacement of the object. If the force is not constant, we can still calculate the work using a graphical technique. If we divide the overall displacement into short segments, D s, the force is nearly constant during each segment. The work done during that segment can be calculated using the previous expression. The total work for the overall displacement is the sum of the work done over each individual segment:

This sum can be determined graphically as the area under the plot of force vs. distance. (If you know calculus, you may recognize this sum as leading to the integral   .)

These equations for work can be easily evaluated using a force sensor and a Motion Detector. In either case, the work-energy theorem relates the work done to the change in energy as

W = ΔPE + ΔKE

where W is the work done, ΔPE is the change in potential energy, and ΔKE the change in kinetic energy.

In this experiment you will investigate the relationship between work, potential energy, and kinetic energy.

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Prelab

1. Lift a book from the floor to the table. Did you do work? To answer this question, consider whether you applied a force parallel to the displacement of the book.

2. What was the average force acting on the book as it was lifted? Could you lift the book with a constant force? Ignore the very beginning and end of the motion in answering the question.

3. Holding one end still, stretch a rubber band. Did you do work on the rubber band? To answer this question, consider whether you applied a force parallel to the displacement of the moving end of the rubber band.

4. Is the force you apply constant when you stretch the rubber band? If not, at what point in the stretch is the force the least. At what point is the force the greatest?

1. Procedure:  Work when the force is constant.

In this part you will measure the work needed to lift an object straight upward at constant speed. The force you apply will balance the weight of the object, and so is constant. The work can be calculated using the displacement and the average force, and also by finding the area under the force vs. distance graph.

1. Equipment Setup: A force sensor is connected to "CH 1" and a motion detector is connected to "DIG/SONIC 1" of the Vernier LabPro interfaced. The range switch on the force sensor is set to 10 N. Computer software for data collection is the LoggerPro software loaded with "WorkEnergy" experiment. Notice three graphs are being used of: distance vs. time, force vs. time, and force vs. distance.

2. Place the Motion Detector on the floor, away from table legs and other obstacles. Be extra cautious during this experiment not to drop weights onto the detector. Hang a 200g mass from the Force Sensor.

3. Hold the Force Sensor and mass about 0.5 m above the Motion Detector and click "Start" to begin data collection. Wait about a second after the clicking sound starts, and then slowly raise the Force Sensor and mass about 0.5 m straight upward. Then hold the sensor and mass still until the data collection stops at 5 s.
Try this a few times till you feel you have a set of data representative of your experiment.

4. Sketch the three graphs into your notebook

5. Examine the distance vs. time and force vs. time graphs by clicking the Examine button . Identify when the weight started to move upward at a constant speed. Record this starting time and height in a data table and on your sketched graph

5. Examine the distance vs. time and force vs. time graphs and identify when the weight stopped moving upward. Record this stopping time and height in a data table and on your sketched graph.

6. Determine the average force exerted while you were lifting the mass. Do this by selecting the portion of the force vs. time graph corresponding to the time you were lifting (refer to the position graph to determine this time interval). Do not include the brief periods when the up motion was starting and stopping. Click the Statistics button, to calculate the average force. Record this value in your data table.

7. On the force vs. distance graph select the region corresponding to the upward motion of the weight. (Click and hold the mouse button at the starting distance, then drag the mouse to the stopping distance and release the button.) Click the Integrate button  to determine the area under the force vs. distance curve during the lift. Record this area in the data table.

Procedure:  Work done to stretch a spring.

In this part you will measure the work needed to stretch a spring. Unlike the work needed to lift a mass, the work done in stretching a spring is not a constant. The work can still be calculated using the area under the force vs. distance graph.

1. With the "WorkEnergy" experiment loaded, change the force scale of the Force/Time and Force/Distance graphs to go from 0 to 0.6N.

2. Attach one end of the spring to a rigid support and the Force Sensor hook to the other end. Rest the Force Sensor on the table with the spring extended but relaxed, so that the spring applies no force to the Force Sensor. Place the Motion Detector about one meter from the Force Sensor, along the line of the spring. Be sure there are no nearby objects to interfere with the distance measurement.

3. Using tape, mark the position of the leading edge of the Force Sensor on the table. The starting point is when the spring is in a relaxed state. Hold the end of the Force Sensor that is nearest the Motion Detector as shown in Figure 3. The Motion Detector will measure the distance to your hand, not the Force Sensor.

4. Click "Start" to begin data collection. Within the limits of the spring, move the Force Sensor and slowly stretch the spring about 50 cm over several seconds. Hold the sensor still until data collection stops. Do not get any closer than 40 cm to the Motion Detector.
Try this a few times till you feel you have a set of data representative of your experiment.

5. Sketch the three graphs into your notebook

6. Examine the distance vs. time and force vs. time graphs by clicking the Examine button . Identify the time when you started to pull on the spring. Record this starting time and distance in a data table and on your sketched graph

7. Examine the distance vs. time and force vs. time graphs and identify the time when you stopped pulling on the spring. Record this stopping time and distance in a data table and on your sketched graph

8. Click the force vs. distance graph, then click the Linear Regression button  to determine the slope of the force vs. distance graph. The slope is the spring constant, k. Record the slope and intercept.

9. The area under the force vs. distance graph is the work done to stretch the spring. How does the work depend on the amount of stretch? On the force vs. distance graph select the region corresponding to the first 10 cm stretch of the spring. (Click and hold the mouse button at the starting distance, then drag the mouse to 10 cm and release the button.) Click the Integrate button  to determine the area under the force vs. distance curve during the stretch and record this area in your data.

10. Now select the portion of the graph corresponding to the first 20 cm of stretch (twice the stretch). Find the work done to stretch the spring 20 cm and record this in your data.

11. Select the portion of the graph corresponding to the maximum stretch you achieved. Find the work done to stretch the spring this far and record in your data.

Procedure:  Work done to accelerate a cart.

In this part you will push on the cart with the Force Sensor to cause the cart to accelerate. The Motion Detector allows you to measure the initial and final velocities; along with the Force Sensor, you can measure the work you do on the cart to accelerate it.

1. With the "WorkEnergy" experiment loaded, change the force scale of the Force/Time and Force/Distance graphs to go from 0 to 0.6N.

2. Determine the mass of the cart and record it in the data table. Place the cart at rest about 1.5 m from the Motion Detector, ready to roll toward the detector.

3. Prepare to gently push the cart toward the Motion Detector using the Force Sensor. Hold the Force Sensor so the force it applies to the cart is parallel to the sensitive axis of the sensor.

4. Click "Start" to begin data collection. When you hear the Motion Detector begin clicking, gently push the cart toward the detector using only the hook of the Force Sensor. The push should last about half a second. Let the cart roll toward the Motion Detector, but catch it before it strikes the detector. Try this a few times till you feel you have a set of data representative of your experiment.

5. Sketch the three graphs into your notebook

6. Examine the distance vs. time and force vs. time graphs by clicking the Examine button . Identify when you started to push the cart. Record this time and distance in the data table.

7. Examine the distance vs. time and force vs. time graphs and identify when you stopped pushing the cart. Record this time and distance in the data table.

8. Determine the velocity of the cart after the push. Use the slope of the distance vs. time graph, which should be a straight line after the push is complete. Record the slope in the data table.

9. From the force vs. distance graph, determine the work you did to accelerate the cart. To do this, select the region corresponding to the push (but no more). Click the Integrate button  to measure the area under the curve. Record the value in the data table.

Analysis

Work when the force is constant

• The work you did in lifting the mass did not change its kinetic energy. The work changed the potential energy of the mass. Calculate the increase in gravitational potential energy using the equation,  ΔPE = mgΔh  where Δh is the distance the mass was raised. Compare this to the work done as calculated by the area under the force vs. distance graph as well as calculated from the average force over the distance. Record your values in the data table. Does the work done on the mass correspond to the change in gravitational potential energy? Should it?

Work done to stretch a spring

• In this part you did work to stretch the spring. The graph of force vs. distance depends on the particular spring you used, but for most springs will be a straight line. This corresponds to Hooke’s law, or F = – kx, where F is the force applied by the spring when it is stretched a distance x. k is the spring constant, measured in N/m. What is the spring constant of the spring? From your graph, does the spring follow Hooke’s law? Do you think that it would always follow Hooke’s law, no matter how far you stretched it? Why is the slope of your graph positive, while Hooke’s law has a minus sign?

• The elastic potential energy stored by a spring is given by ΔPE = ½ kx², where x is the distance. Compare the work you measured to stretch the spring to 10 cm, 20 cm, and the maximum stretch to the stored potential energy predicted by this expression. Should they be similar? Note: Use consistent units. Record your values in the data table.

Work done to accelerate a cart

• In this part, you did work to accelerate the cart. In this case the work went to changing the kinetic energy. Since no spring was involved and the cart moved along a level surface, there is no change in potential energy. How does the work you did compare to the change in kinetic energy? Here, since the initial velocity is zero, ΔKE = ½ mv ² where m is the total mass of the cart and any added weights, and v is the final velocity. Record your values in the data table.

Further

1. Show that one N•m is equal to one J.

2. Start with a stretched spring and let the spring do work on the cart by accelerating it toward the fixed point. Use the Motion Detector to determine the speed of the cart when the spring reaches the relaxed position. Calculate the kinetic energy of the cart at this point and compare it to the work measured in Part II. Discuss the results.

3. Repeat constant force experiment, but vary the speed of your hand as you lift the mass. The force vs. time graph should be irregular. Will the force vs. distance graph change? Or will it continue to correspond to mg·Δh?

4. Repeat cart experiment, but start with the cart moving away from the detector. Pushing only with the tip of the Force Sensor, gently stop the cart and send it back toward the detector. Compare the work done on the cart to the change in kinetic energy, taking into account the initial velocity of the cart.