physics lab

NORTHERN ILLINOIS UNIVERSITY PHYSICS DEPARTMENT Atwood Machine Apparatus The Atwood machine is a simple device consisting of two unequal masses that are connected by a cord run over a pulley. The larger mass is suspended above the smaller mass , and sits upon a platform that is equipped with a mechanical trip release. Upon release, the heavier mass accelerates down to a platform a measured distance below. Each group will measure the time required for to move from the upper to the lower platform. Theory The masses in the Atwood machine are subject to a number of forces, all of which are constant. The net constant force will produce a constant acceleration. We know that a co nstant acceleration will cause an object (starting from rest) to move a distance equal to . If we measure the distance and time we can solve for the acceleration. (1) A perfect Atwood machine is a massless and frictionless pulley. In such a case the force s acting on the heavy mass and light mass are (the positive y-direction is defined as downward) : (2) (3) where is the tension in the string , and the acceleration of each mass is equal (but in opposite directions) . Subtracting Eq. (3) from Eq. (2) gives: (4) where positive acceleration has been defined as in the downward direction. Physics 253 – Basic Mechanics Read Giancoli: Chapter 5 2 m 1 m 0 y y y   t 2 m a 2 12 y at 2 2 y a t   2 m 1 m 22 m g T m a  11m g T m a    T     2 1 1 2 m m g m m a    Our real Atwood machine has a pulley having a nonzero mass that resists acceleration (that is, it has rotational inertia) . The mass of the pulley is being accelerated by the pull of the string at its rim. We will learn later in the course that a rotating disk provides an additional inertia at its rim equal to half of its mass: . For this experiment w e can use as if it were an additional mass to be accelerated with and : . How should we deal with the frictional force? If it acts like kinetic frict ion it should be a constant force . If we include this as an additional force opposing the downward motion of , then: . With these modifications, the equation for Newton’s second law becomes: (5) In our experiment we will determine the acceleration compared to the mass difference . If we rearrange Eq. 2 to solve for (6) The equation for a straight line is , ( is the vertical axis , is the horizontal axis ) where is the slope and is the -intercept. Eq. 4 has a similar form. If we plot (vertical axis ) vs (horizontal axis ) the result should be a straight line with a slope equal to , and a -intercept at . Data Collection string . Separately measure and record in your lab notebook the mass and the uncertainty in the mass of each weight ( aluminum cylinder s and disks ) to be hung on the strings (and write the weight on the small piece of tape) . Exercise caution: though the weights look alike, they do not have precisel y the same mass (thus the reason for the labels) . (1) Read and record the mass of the pulley, m p , from the side of the apparatus. DO NOT REMOVE THE PULLEY FROM THE APPARATUS. (2) Put a small piece of scotch tape (to write on) on each weight to be hung from the p m 2/p m 2/p m 1 m 2 m     1 2 1 2 2/p m m a m m m a     f 2 m     2 1 2 1 m m g m m g f       2 1 1 2 2 p m m m g f m m a        a   21 mm  a       21 1 2 1 2 22 pp gf a m m mm m m m m         y mx b  y x m b y a   21 mm    12 2 p m g m m  y   12 2 p m f m m   (4) On one aluminum cylinder, place one 5 gram mass, three 2 gr am masses, and one 1 and any aluminum disks placed on them). Also, open up Excel and make a table acceleration of the mean, the average time and and enter them into you r Excel table and lab notebook . (6) Keeping the total mass the same (roughly 12 grams) , exchange weight(s) between the cylinders such that the mass difference is 10 grams instead of the current 12 grams. Record in your lab notebook your new values for , and the mass difference in grams . Repeat step (5) . (7) Repeat step (5 ) for mass differences of 8, 6, and 4 grams. In the end you should have a total of five sets of data, with mass differences of about 12, 10, 8, 6, and 4 grams, but the total mass is always the same (roughly 12 grams). Thus the denominator in Eq. 4 is a constant (it does not vary between measurements ). (8) Calculate the acceleration and its uncertainty using Eq . 1 above (use the method of propagation errors to calculate the uncertainty). Calculate the relative error (that is, the uncertainty of divided by ). En ter these value s in your Excel spreadsheet and lab notebook. A relative error of 1% is considered a very good measurement. xx  in Excel]).  \f x SQRT x  and 2 2^   (do the standard deviation calculation using Excel – [Note: mt , time t , and the standard error a a , relative error in the acceleration amm  , acceleration a , uncertainty in 21 having 6 columns for: the mass difference  \fmm  for the first data set in grams (remember, 1 m and 2 m are the total masses of the cylinder 21 string. Record in your lab notebook 1 m , 2 m and the mass difference  \f gram mass (making a total of roughly 12 grams). Let 1 m be the total mass hanging on one side of the string and 2 m be the total mass hanging on the other side of the the lower photogate and record this value. ( Gently catch the falling mass with your hand after it passes through the lower photogate — this will prevent the masses from falling off the Atwood machine ). This should be repeated for a total of 5 trials without changing the masses. Record each trial in your lab notebook and calculate (5) Using the Logger Pro software, time the fall of the heavier weight from the upper to (3) Adjust the two photogates to be about 70 cm apart and first photogate should be right next to launch pad. Make certain that the weights do not make contact with the photogates and pass through the motion sensor. Measure and record the distance (and its uncertainty) from the upper to the lower photogate . t t   12mm    21 mm  1 m 2 m   21 mm    21 mm    12mm  a a a a Remember: Take a picture of your experimental setup with your smart phone. The calculations in the Analysis section are to be finished before lab ends. You must show your final result (in Part (6)) to your TA before you leave the lab. Analysis (1) In the lab , p lot using Excel the acceleration in cm/sec 2 versus the mass difference in grams for th e five data sets. Note that acceleration should be the vertical axis. Include the error b ars by using your unc ertainty calculations in step (8 ) of the Data Collection section. (2) In the lab , using Excel, fit the data with a straight line (showing the fitting equation (4) Find the frictional force on the pulley by calculating the product of the -intercept with the total mass (see Eq. 4). Convert the value of the force to Newtons. (5) Find the gravitational acceleration by calculating the product of the slope with the total mass (see Eq. 4). Convert the value of the acceleration to m/sec 2. (6) Compute a percent error between your average and 9. 8 m/sec 2. Show your result to your TA before you leave the lab. A 1% error would be very good. A 10% error is what one would expect for the instruments used in this lab. on the plot). (3) In the lab , write the y -intercept and slope values in your logbook. a   21 mm  f y g g 100 % Experimental AcceptedValue Error AcceptedValue  