1Experiment 2 Uniform Acceleration in One Dimension Preparation Prepare for this week's experiment by reading about velocity and acceleration. Review definite integrals in your calculus book. Principles Objects dropped near the surface of the earth will accelerate downwards. While the value for the acceleration due to gravity, g, varies from place to place on the planet, all objects dropped at one place will accelerate downward at the same rate. In today's experiment you will take data that will enable you to calculate a value for g. Suppose that you could drop an object and make a record of its position at specific times. How could you use that data to find the value for g? Consider the motion of a particle traveling in one dimension that is under going constant acceleration. We can choose an instant in time and call this time t = 0. Its position is xo, its velocity is vo and its acceleration is a. xo and vo are fixed values which may or may not be equal to zero. At some later time, t, the particle has position, x, and velocity, v, which are both functions of time. From your class work and readings you know that the position of this particle is given by 21200x=x +vt+ at . If you plot the position as a function of time the graph will be a parabola. It will open upward if the acceleration is positive. At any point on the line the tangent or instantaneous slope will equal dx=vdt. If you differentiate the equation for position you can see that the velocity equals 0v=v +at. If you plot the velocity as a function of time you will get a straight line. The slope of that line will equal dv=adt. Suppose you dropped an object and marked its position at a number of times so that the time intervals are all equal and fairly small. If you selected a starting point and measured the position2The slope between two close points gives you the average velocity over the time interval. The average velocity equals the instantaneous velocity at the midmark time.Mark time #1 Mid-mark time Mark time #2 Time (1/60 sec)of the falling object relative to that point you could plot the positions as a function of time. The graph you get would not be perfectly smooth, but it would come close to the ideal curve. You wouldn't be able to find the instantaneous slope at any point on this curve, but you could find the average slope between two adjacent points. Designate each position as xi, as i goes from 1 to n, where n is the total number of points. Do the same for the times at which the marks were made; call them ti. The average slope equals the average velocity of the object over that time interval such that i+1 iii+1 ix-xv=t-t. Since the object is accelerating at a constant rate, some of the time it will be going faster than the average velocity and some of the time it will be going slower. At the midpoint of the time interval, ()12ii+1t+t , the average velocity and the instantaneous velocity will be equal. This value ()12ii+1t+t is called the mid mark time. You would now know how fast the object was moving at specific times. If you plot these velocities as a function of time the slope of the line will give a value for the acceleration. The inverse of the time difference is the frequency, f, of the sparker i+1 i1f=t-t, so vi = f (xi+1 –xi) Equipment 1 length of heat-sensitive tape 1 2-meter stick masking tape Procedure The free fall apparatus has an electromagnet to hold the object until a record of its fall is to be made. When the switch to the coils is opened the object will fall freely into a cup at the bottom of the apparatus. The object, which is insulated except for a metal ring around its center of mass, will fall between two stretched wires which are connected to a high voltage spark generator which produces pulses 60 times per second. A strip of heat sensitive paper is stretched over one of the wires. As the object falls a spark will jump from one wire, through the conducting ring, Position (cm)3. . . . . . . . . through the paper to the other wire. This produces a record of the position of the object at 1/60 second intervals. The lab instructor will demonstrate the apparatus if it is available and give each group of students a prepared tape. Because the apparatus is potentially dangerous, you will not be allowed to make your own tape. Use centimeters for your distances. Measure positions to the nearest millimeter and estimate hundredths of centimeters. 1. Find one or two people to work with. 2. Get a prepared tape from your instructor. Stretch it out on the table so that the distances between the spots are increasing from left to right and secure the ends with masking tape. Find the part of the tape where the marks are about 2 cm apart. Go down the tape circling every mark except for the last one. Check the tape to be sure that no marks are missing. 3 Take data in ink and write it directly into the notebook. Write down the names of you lab partner or partners. Write down your spark frequency. Start a column labeled "Position". Put the correct unit in parentheses. There is no need to number the positions or to write anything else on your data page. 4. Lay a two meter stick on its side and place it on the tape so the first circled mark is at the 10.00 cm mark on the meter stick. This corresponds with position xo at an arbitrary t=0. Without moving the meter stick, read the position of each subsequent mark to the nearest hundredth of a centimeter and record it directly into your notebook. This means that you will have to estimate the values between the mm marks. If your measurements all seem to end in 0 or 5, you are not measuring correctly; start again and measure more carefully. 5. Do not destroy your tape until you've completed the analysis. You may need to refer to it again. Data The data for this experiment consists of the spark frequency (60 Hz) and the positions of the marks. Be sure you have your instructor check and sign your data before you leave for the day. Analysis You must take data and do most of the analysis correctly to get credit for participating in the lab for this week. Do as much of the analysis as you can before you leave for the day. You should at least finish the spreadsheet before you go. Have your instructor check it when he or she initials your data.4The mark times for this experiment are 0, 1/60 s, 2/60 s, etc. This is very