PHY 221 Lab 7 Work and Energy

Transcription

1 PHY 221 Lab 7 Work and Energy Name: Partners: Goals: Before coming to lab, please read this packet and do the prelab on page 13 of this handout. Note: originally, Lab 7 was momentum and collisions. The order was swapped to sync with PHY211. While F = ma may be the most important equation you learn in your first physics course, the concept of energy may be the most important idea. Energy is a quantity characterizing a physical system that keeps the same value ("is conserved", in physics terminology) throughout a variety of transformations. In this lab, you ll examine a few examples where the concept of energy helps you see some surprising regularities in mechanics problems. You ll start by working with concepts called work and kinetic energy. Then, you will investigate gravitational potential energy. Please do PRELAB on page 13 after you have read this lab writeup. Materials: PC with LabQuest computer interface for measuring instruments Motion Sensor Force Probe either tethered or Wireless Two PASCO carts on aluminum tracks Two rectangular weights for the cart Balance to measure masses Light spring Mounting rods and fixtures 1

2 Activity: 1. Work and kinetic energy for moving cart Work is something that can be measured with the instruments you ve become expert in using this semester in PHY 221. Since work is the integral of the force applied to an object over the distance of force application (integral of F ds ), you just need to measure the force applied to an object (F) and the distance over which the object moves (s), and then be able to numerically evaluate the integral. These measurements you ve already learned to make with the force probe and the sonic ranger. It turns out that the Logger Pro software is also capable of evaluating this integral too. Start out by using a setup similar to one of those from a prior lab, namely, a cart with a force probe bolted on top. Set the cart with the force probe on the track, with the hook of the force probe facing away from the sonic ranger. Wake up Logger Pro and make sure that the sonic ranger can detect the cart all along the track. Attach a light spring to the force probe hook. You will be pulling on the other end of this spring to exert force on the cart that will vary smoothly with time. Zero the force probe. Grab the spring, without applying any force to the hook yet. The cart should be at rest. Start collecting data. Stretch the spring applying a force to the cart. The cart will start moving. After about half way along the track release the spring from your hand and let the cart move freely. Practice recording clean data. Check that the cart rolls freely. The recorded force should be initially zero, then it assumes some positive values, finally it becomes zero again when you let the cart go freely. Once the cart moves freely it should have constant velocity that will show up as linear dependence of the cart s position on time. PRINT your best effort of Position vs. Time, Force vs. Time, and Force vs. Position. Indicate the zero values of the force by drawing dashed horizontal lines across the last two graphs. To calculate the work done by you on the cart, select a range (click and drag) on the Force vs. Position graph from the initial cart position to the distance when it became free (i.e. when the force became zero again). Go to Analyze menu and select Integral. Make sure that the selection bars don t disappear when you do it. You will need to redo the procedure if they do. Graphically, the integral F ds, is equal to the area under the 2

3 Force vs. Distance curve. The program will highlight that area, and the pop-up box will give you a numerical value of the integral. Shade the integrated area in the report graph as well. Units of the integral are Newton times meter (N m) which is also called Joule (J). Copy this value of work to the first row and first column of the table that you can find below. To see why this integral (called work) is useful, let us define kinetic energy of an object at motion as K = m v 2 / 2. While work describes the interaction with the cart along the integrated distance range, kinetic energy of the cart is a quantity that we can assign to the cart at any moment of time. Since the initial velocity is zero (cart at rest), the initial kinetic energy is zero too, Ki = 0. Let us calculate kinetic energy of the cart after the force became zero again, Kf. Stretch a selection range on the Position vs. Time graph to cover the linear part of the graph right after the force became zero (for better accuracy don t stretch it too far). Indicate the selected time range with vertical bars on the graph in your report. Go to Analyze menu and select Linear Fit. If you have done this correctly the superimposed straight line on the graph will be tangent to the linear part of the graph. Slope of the line (given in the box as m = ) is the cart velocity. Alternatively you can use the average Velocity vs. Time graph in the corresponding range. Store the measured velocity in the table. To calculate kinetic energy from velocity (v) we need to know the mass. Measure the mass of the cart together with the force probe using a balance. For the record, list this value here: m = kg From the measured velocity and mass calculate the final kinetic energy of the cart, Kf. Units of kinetic energy are kg m 2 /s 2. Since Newton is equivalent to kg m/s 2 (Newton s 2 nd Law!), units of kinetic energy are N m = J; the same as for work. Store this kinetic energy in the table, in the column denoted as change of kinetic energy for the same interval for which we calculated the work, K = Kf Ki = Kf 0 = Kf. Repeat the whole experiment. Try to give the cart as large a velocity as you can before cutting it loose. Don t forget to zero the force probe before every measurement. Store results in the second row of the table. Take two more measurements. Each time try to apply force differently. Apply it with different strength and different dependence on time. Try to cover a wide range of final velocities. 3

4 Experiment Work - W (J) Final velocity - vf (m/s) Change of kinetic energy K (J) Use LoggerPro to construct a graph of K vs. W. PRINT the graph and attach it to this report. Do you see a relation between K and W from your graph? If so, what is it? 4

5 2. Can work can be negative? Disconnect the spring from the force probe hook. Put two rectangular weights on top of the cart to increase its mass by 1 kg (each weight is about 0.5 kg). Check that sonic ranger detects the cart all along the track. Zero the force probe. Start collecting data. Give the cart a strong push and let it go without allowing the weights to fall off (don t grab the force probe hook we are not interested in recording the force you applied here). After a second or two, try to slow it down until it stops (but don t make it go back) by applying a gently force to the probe hook opposing the cart motion. PRINT graphs of Position vs. Time, Force vs. Time, and Force vs. Distance and attach to report. What is the sign of the recorded force during the time when you were slowing the cart down? What is the meaning of this sign? Integrate the Force vs. Position graph in the interval where you were slowing the cart down (force should be zero at the beginning and at the end of this interval). Store the calculated work in the table below, together with its sign. Shade the integrated area on your graph. Measure the initial velocity of the cart before you applied the force to slow it down. Indicated the range selected for velocity calculation on your graph (select this range right before you started slowing the cart down). Calculate change of kinetic energy during the time you were applying force to the probe hook. Give it a proper sign. Put these results in the table. Repeat the experiment twice more, giving the cart different initial velocities. 5

6 Experiment Work - W (J) Initial velocity - vi (m/s) Change of kinetic energy K (J) Make a graph of K vs. W again. Does your graph roughly agree with the work kinetic energy theorem, W = K? Even though you may think you worked hard to stop the cart, this work turned out to be negative. Thus, you took kinetic energy away from the cart. The cart spent its energy to push your finger. The cart did positive work on your finger. In our experiments we were neglecting friction of the cart on the track, since the frictional force is small in our set-up. What is the sign of work done by the frictional force on the moving cart? Why? 6

7 If friction were large, what role would it play in changing kinetic energy? Should large friction be included in W = K relation? How? In the first activity we were starting the cart from rest. We know that before the cart started to move there had to be some static friction opposing the force. Did static friction do any work? Explain? 3. Gravitational potential energy and conservation of total mechanical energy From the previous activities, we have learned that work can be stored in motion of an object. Amount of the work stored in motion is quantified by kinetic energy. Kinetic energy is not the only way to store work. Work can also be stored in a separation of two masses. They attract each other by gravitational force. This gravitational force can do work. When an object falls down, attracted by the Earth, it gains kinetic energy from the work done by gravity. Thus, an object raised above the ground has a potential for doing work. We can say that it has some potential energy stored in it. This potential energy turns into kinetic energy. Conversely, if you throw an object up, its initial kinetic energy will be turning into potential energy on the way up. 7

8 From previous labs we have also learnt that kinetic friction robs a portion of the kinetic energy, slowing the cart s velocity as it rolls down an incline, so we need to measure its effect as well in our experiment. Change of potential energy can often be calculated more easily than work done by the associated force (here gravity), which makes this concept very useful. In this part of the lab, you should remove the force probe, weights and spring from the cart (or just use a cart that has nothing attached to it). Tip the track up at an angle so that the sonic ranger is at the high end, using the post and clamp setup to hold it in place (initially set the ranger about 30 cm above the table). The thing to measure here is a simple downward rolling motion of the cart, after you release it from a particular height. Pick a convenient spot on the track from which to release the cart, and measure the height of the track at the midpoint of the cart. Now, practice gently releasing the cart and letting it roll until it reaches near the end of the track. Make sure you can recognize in your data the moment when the cart hits the table upon reaching the bottom end of the track. (One team member s job is to stop the cart from falling onto the floor.) Be sure you are in Logger Pro using as before the File from PHY221Fall 2011/ Work and energy. When you have obtained clean data, start filling the DATA TABLE in Logger Pro which is similar to that on page 10. Fill in the angle of the track θ (in degrees), the mass of the cart, the initial height of the cart, its acceleration and the cart s velocity at the end of the track as measured with the Sonic Ranger and recorded with Logger Pro. All values will have to be entered for every experimental trial even though some are the same in every trial. Release the cart from two other heights along the sloping track. Enter these values into the Logger Pro DATA TABLE. Logger Pro will automatically calculate the rest of the Table. Be sure you understand each calculation the computer has done for you. Ask the instructor if you have questions. Now, adjust the slope of the track to a substantially different value. Fill in the lower section of the table with a new set of final velocities from motions that start at several different heights, including some of the same height as in the previous set. PRINT the DATA table showing both your data and the calculations the computer did. Analysis of data in the table. Compare two final velocities obtained with the track at different angles, but the cart released from the same height. How do they compare? Pay particular attention to the changes of kinetic energy between initial and final positions of the cart on the track (ΔK) 8

9 Only use the friction correction if it is greater than 1. Why? Does the correction make sense in view of your previous studies? Explain. Have the computer draw a graph of K vs. height. Have the computer draw a graph of K (corrected for friction) vs. height on the same graph as above. PRINT the combined graph What kind of relationships do you see in the graphs? 9

10 Experiment # Height - h (m) Average Acceleration (AA) (m/s 2 ) Final velocity (v f ) (m/s) Friction correction 9.8(m/s 2 ) sin /AA Friction Corrected Final velocity (v fc ) (m/s) Measured Change of kinetic energy K (J) Friction Corrected Change of kinetic energy Kcor Fraction of kinetic energy lost to friction Mass of cart = 1 (J) Initial Angle of the track with respect to the horizontal, = degrees Different slope of the track, = Note: The Friction corrected Final velocity (v fc ) = the Final velocity (v f ) the Friction correction. Fraction of Kinetic energy lost to friction = [Corrected K.E. Kcor) (Uncorrected K.E. K)]/ Kcor 10

11 Do points coming from different track inclinations fall on the same curve, with and without consideration of friction? What conclusion can you draw from these results? Gravitational potential energy of an object above the ground is given by U=mg h. We can also define total mechanical energy as the sum of potential and kinetic energies E=U+K. If we neglect frictional forces, the total mechanical energy should remain constant with time (i.e. be conserved). Discuss how definition of U, together with the conservation of total energy, can explain the dependence of K on h that you observe in your experiments? What is slope of U versus corrected K? How close is this to your expectation on the assumption of no friction? 11

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13 Lab 9 Prelab Re: Conservation of Energy What would you expect for Activity 3? (Describe accurately in detail your expectations for all the relationships you are going to measure and plot.) 13