Uniformly Accelerated Motion In One Dimension 1 Objective To experimentally study 1-dimension motion with uniform acceleration 2 Overview The main idea of this experiment is to study the motion of a cart on a (nearly) frictionless inclined track. and obtain an experimental value of its acceleration. Then, from its accelers ation, obtain an experimental value of the acceleration due to gravity, ,9, and compare it to the theoretical value of 9.8 111/52. Refer to Figure l, where a cart of mass m rolls down a frictionless track that makes an angle 6 relative to the horizontal plane. Let :1 and :2 be two positions of interest along the track. As you will learn later on in class, the cart experiences an uniform acceleration, a that points along the track (here, chosen as the positive :- aicis) with a :rcomponent, 41,. = g sin'. (1) Note: The acceleration will still be uniform, though with magnitude less than 9 sinG in the presence of non-zero friction, as long as the friction is uniform. Again, you will learn this later on during a discussion on Newton's laws of motion. Recall from kinematics that the Icomponent of the instantaneous velocities, U12. and I)\" at positions, I1 and r3, are related by the following equation: vi. = 12% + 2:141; 11) (2) Further, if h and :2, are the time instants at which the cart is at positions, :1 and :2, respectively, then. 1 I: = 11 + leliz i1) + i023: f1)? (3) In Part 1 of the experiment, you will employ two photo gates positioned at 11 and I; that For use by the physics department, ()hlone College, Fremont. CA. Lab Instructions V0.0 June 18, 2018 work independently. The photo gates will measure the time, An and dig, it takes the cart to clear each of the individual photo gates. This will help you estimate U1, and v21. Then, from Equation 2. you can derive an experimental value of of. In Part 2 of the experiment, the two photo gates positioned at :1 and I; are oonnected in tandem, which will help you measure the time of travel of the cart from gate 1 to gate 2, i.e.. A: : (t3 :1). Then, from Equation 3, you can derive an experimental value of ax. Once you have an experimental value of a, in Part 1 or Part 2, you can obtain an experi mental value of g from Equation 1, and oompare it to the theoretical value of 9.8 mfsz. Figure 1: Overview 3 Apparatus 1. Inclined tack that is about 2 m long. 2. A cart that rolls freely on the track. 3. Two primary photo gate units. These have built-in displays. 4. One secondary photo gate. This does not have a display unit. It plugs into a primary gate, and is utilized in Part 2 of the experiment. 5. A stop that clamps on one end of the track A book or some other object to prop up the track. Masking tape. 907\"?\" An index card or similar piece of stiff paper to make a ag taped to the cart. For use by the physics department, Ohlone College, Fremont. CA. Lah Instructions V0.0 .Iune 18, 2018 Figure 3: Experimental Setup 4 Precaution - KEEP THE ANGLE 0F INCLINATION OF THE TRACK SMALL (5 5 degrees)I SO THAT THE CART ROLLS ONLY SLOWLY AT ALL TIMES. - MAKE SURE THAT THE CART DOES NOT ROLLOFF THE TRACK AND CRASH INTO SOMETHING. 5 Part. 1: Use two primary photo gates 1. Prop up the track at one end with a hook or thin box, etc., to create a ramp. Rest the other end on the table. 2. Clamp the stop on the end of the track that rests on the table. If you don't have a stop, use some other object that will gently bring the cart to a stop. 3. Use rightangle trigonometry to create the angle 0 to be close to 2 or 5 degrees. See Figure 1 . Note: Don't simply use a protractor to measure the angle. It is not precise enough. 4. Place the two (primary) photo gate units along the track, separated by at least 1 m. Record their positions as :1 and :2. 5. Mount a aga piece of still paper, like an index. card or cardboard of known Width in to one side of the cart so that it interrupts the photo gate IR. beam, While the cart clears the photo gate. Fhr use by the physics department, Ohlone College, Fremont, CA. Lah Instructions V0.0 June 18, 2013 6. In the analysis section your lab report, discuss the pros and cons of choosing a wide or a narrow ag, i.e., large or small or 7. Set the photo gates to gait: mode and sensitivity to 0.1 ms. Position the cart in front of gate 1 and record it as In. Release the cart from rat or give it a very gentle push. In pulse mode, the gates measure the time, A11 and At2, it talces the cart to clear each of the individual photo gates. The gates do so by measuring the duration for which the ag on the cart interrupts the photo gate IR beam. 8. "y it out a few times, to get a feel for the process and see if the At,- measurements make sense. 9. Record 6, w,m,1n, 11,12 in Table 1. 10. Now, do your rst real run. Make sure to carefully record the position 1'\" from which you released the cart, as you will have to start from the same ID for all subsequent runs. Record At, and Atz in Table 2. 11. Repeat the process for at least N : 5 runs, releasing the cart each time from the same position, In. mwrgrlmg Table 2: Data for Part I Obsenration no. 52 5.1 Calculations l. Utilize w and At, to compute the average velocity of the cart as it clears gate i. This gives an estimate of the required instantaneous velocity v\". Record the values in Table 2. 2. Born the data of the N runs, compute an average value for U1: and 2123. Utilize Equation 2 to derive the experimental value of a2 and then, an experimental value of g from Equation 1. Record your results in Table 3' Lab Instructions V0.0 June 18, 2018 5.2 E'ect of m and 6' 1. Place a mass of about 250 g on top of the cart and repeat the experiment and calcu- lations. 2. Change 5 to the other value out of: n 2 or n 5 degrers and repeat the experiment and calculations. 5.3 Error analysis and discussion 1. What is the tradeho' in choosing the size a: of the flag? 2. Discuss possible sources of error that contribute to the error in the experimental value of g? 3. Do the values of :12 and g depend upon m? 4. Do the value of (1,; and g depend upon I9? Iah D 3. Be' 1m. 2 them \"Expos? at 9171/5 6 Part 2: Use one primary and one secondary photo gate, connected to work in tandem 1. Follow the procedure in Part 1 up to step 3. 2. Choose one primary gate (gate 1), and one secondary gate (gate 2). Connect the photo gates in tandem by connecting the triggering cable from gate 2 into gate 1. 3. Place two photo gate units along the track, separated by about 20 cm. Record their positions as :1 and :9. 4. Position the cart such that front end (leading edge) of the ag is ccsctly at gate 1 (primary gate), so that the photo gate timer starts running as soon as you release the cart. 5. Set the photo gates to pulse mode and sensitivity to 0.1 ms. The primary gate timer will stop running running as soon as the ag reaches gate 2 (the secondary gate). So, the timer new measures the travel time, At, from gate 1 to gate 2. 5. Try it out a few times, to get a feel for the process and see if the At measurements make sense. Make sure to gently release the cart nm rest, (don't push), ght at the location of gate 1. For use by the physics department, Ohlone College, Fremont, CA. Lab Instructions V0.0 June 18, 2018 7. Now, do your rst real run. Record the :1, r? and At in Table 4. 8. Repeat the process, changing 1;, so that the gates are separated by [20 em, 40 em,60 em,80 cm,100 cm], a total of at least N = 5 runs. . 6. 1 Calculations 1. Sinoe, on : 0 (cart released from rest) and At : (t2 t1), Equation 3. simplies to 1 12 = 2:1 + EozAtz. (4) 2. You need to utilize Equation 4 to obtain or from a plot of your data. For accurate results and for convenience, let's convert Equation 4 in to a linear equation by taking the natural log on both sides. Thus, we get ln(r2 r1) : ln(%a,) + 2111 At. (5) 3. Now plot ln(zg :1) versus LnAt. Draw a best t straight line through your data points, and then obtain 412: from the intercept, and then, an experimental value of g from Equation 1. Record your results in Table 5 4. Change 9 to the other value out of: m 2 or :3 5 degrees, and repeat the experiment and calculations. Table 4: Dat for art Observation no. 1'1 3:2 At 1 N Table 5' Results for Part 2 a: 9 m/s2 % error = (41:31 x 100 6.2 Error analysis and discussion 1. Discuss possible sources of error that contribute to the error in the experimental value of g? 6.2 Error analysis and discussion 1. Discuss possible sources of error that contribute to the error in the experimental value of g? 2. Compare the results of Part 2 with those of Part 1. Which are more reliable? For use by the physics department, Ohlone College, Fremont, CA. Lab Instructions VO.0 June 18, 2018 7 Conclusion Briefly summarize your findings