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Download LAB 3: FORCE AND ACCELERATION Study of Newton`s Second
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Brown University Physics Department Physics 0030 Lab 3 LAB 3: FORCE AND ACCELERATION Study of Newton’s Second Law of Motion An object which is free to move horizontally without friction is subjected to a series of known forces and its acceleration is measured for each force. The measured accelerations are to be plotted graphically as a function of the applied force. REFERENCES: Physics 0030 Guide to Laboratory Measurements; A study of Free Fall Velocities; Kestin and Tauck, University Physics Vol. 1, Chapter 4 (section 4-6). BASIS OF THE EXPERIMENT A linear air track provides a nearly frictionless path for a glider resting on it (Fig. 1). The glider is pulled by a string which passes over a pulley at the end of the track, to a hanger to which known weights can be attached. These weights with the hanger provide the variable applied force. Glider Pulley Track Air from Blower Weights FIGURE 1 Suppose we let: mh = mass hanging on the string (including weight hanger) mt = mass on track (glider plus any weights resting on it) M = m h mt F = total applied force = mh g 131022 1 Brown University Physics Department Physics 0030 Lab 3 T = tension in string (acting horizontally toward pulley, on mt , and acting vertically, upward, on mh ) a = acceleration of mt (horizontal, toward pulley), and also a = acceleration of mh (vertical, downward) Note that mt and mh must move together – that is with the same velocity and acceleration – as long as the string connecting them remains the same length throughout their motion. Fig. 2 shows the forces acting on mt and mh . The equations governing the motion are: (1) mt a T (2) mh a F T Pulley mt T a Adding these and substituting T mh (mt mh )a F gives Ma F which can be written F=mhg (3) a F /M FIG. 2 Equation (3) is what we wish to investigate. It expresses Newton's Second Law of Motion, as applied to the composite system, consisting of glider + weights + hanger + connecting string, while Equations (1) and (2) are expressions of Newton's Second Law applied to mt and mh separately. It is T which actually pulls the glider, and to find T from F we would need to make use of the Second Law as expressed in Eq. (2). However, without using the second law we know that M mh mt is the mass of the composite system and F mh g is the only force applied to it; T is an internal force within the two-body system, not a force applied to it. For the two-body system we know the left- and right-hand sides of Eq. (3) independently, once we measure the acceleration a and the masses, and so can test the equation. 131022 2 Brown University Physics Department Physics 0030 Lab 3 PLAN OF THE EXPERIMENT In the experiment, M remains constant while movable pieces of mass (loose weights) are transferred from glider to hanging support. Each such transfer decreases mt and adds to mh (and therefore increases F without changing M). For each value of F, a is constant during the motion. Since M remains constant, Eq. (3) becomes the expression of a linear relationship between F and a in this experiment. It predicts that if each acceleration a is plotted as a function of the corresponding force F mh g , the observed values will lie on a straight line passing through the origin, with a slope equal to 1/M. The principle of the acceleration measurement is identical to that used in the free fall experiment (Reference 2). Here it is the glider's acceleration that is measured, so to base it on photobridge measurements, the bridges are mounted on a rod suspended horizontally over, and parallel to, the track (Fig. 3). The acceleration is constant (one of our basic assumptions) for a given value of masses because it derives from the gravitational force acting on those masses; it is not of course, equal to the constant gravitational acceleration of free fall at any time. Backstop Glider L STOP STOP C START U TUC TUL S dUL FIG. 3 The glider rest position ( x 0, t 0, v 0) is at Z. To make the correspondence with Reference 2 clearer, call the photobridge nearest to Z the U (for upstream) photobridge. Call the middle photobridge the C photobridge, (it was the M photobridge in Experiment 1) and the L photobridge is the last photobridge, nearest the pulley. If the glider leaves a fixed point on the track, moving to the right, it will interrupt each of the beams in order. 131022 3 Brown University Physics Department Physics 0030 Lab 3 The upstream (U) interruption starts the two timers. The interruptions at C and L turn off the UC and UL timers respectively. The C photobridge position is adjusted until the UC time reading is one-half the UL time reading. The rest of the acceleration measurement involves measuring distances. These measurements are made readily using the metric scale that is permanently mounted on the track. Initial position and photobridge positions require passing a perpendicular line in space down to the track scale. The position of the C photobeam (at the time midpoint between U and L) relative to the leading edge at Z of the resting glider is the distance S in this experiment, at which we find the instantaneous velocity. The value of that instantatenous velocity v(tc), at the time midpoint between U and L, is equal to the average velocity of the glider in moving from the U to the L photobeam; it is therefore simply the distance UL between these two beams divided by the time registered on the UL timer. With these two numbers, S and v(t c ) , we can then deduce the acceleration of the glider from v(t c ) v (t u , t L ) DISTANCE UL , and TUL a [v 2 (t c )]2 / 2S , [Alternatively, we might have left the C photobeam at an arbitrary position between U and L and determined the acceleration as in Exp 2.] PROCEDURE Several such acceleration measurements will be made, for different masses mh but constant total mass M. Three small weights are provided, which can be distributed between glider and hanging support in four different ways (number of weights on hanger support = 0, 1, 2, 3). For each of the four distributions of weights you should measure the system's acceleration (that is, the glider's acceleration), at least twice. First determine the total mass of the system by weighing glider, weight-hanger and the three free weights, all together. This total mass (M) remains constant. You will also need to measure the hanger weight and the individual weights to determine mh for various arrangements. To Guard Against Some Potential Problems in Setting up Your System: 131022 4 Brown University Physics Department Physics 0030 Lab 3 Practice starting and stopping the glider to get a feel for the speeds involved and the techniques required. Practice releasing the glider from rest (starting at the backstop position and letting go without imparting any initial velocity). Pick a U photocell position near to the rest position, and an L position near the pulley, note the measured transit time from U to L and repeat several times. If the transit times are not essentially the same, you are giving the glider some initial velocity in the process of releasing it. With all three movable masses loaded on the hanging support (maximum applied force), determine how much room you need to stop the glider near the pulley end, without allowing it to bump into the stop at the end of the track, and without jamming the glider down onto the track. Place the L photobridge to give yourself this room. Record the U and L positions, and leave them there throughout the experiment. With all three movable masses loaded on the hanger, determine the half-time point C of the glider passing through the photobridge array. Do this at least twice before changing the weights, and calculate the corresponding values of acceleration; if there is inconsistency make further adjustments and measurements until the inconsistency is eliminated and you have at least 2 runs in reasonable agreement. (Note the potential problems described in the preceding two paragraphs.) Repeat the last procedure, including at least two runs, with one of the masses transferred to the glider. Repeat again for two masses transferred, then for all three. Remember, you need to specify mh for each mass arrangement. NOTE: It is best for one to start with four loose weights on the hanger initially. This ensures that on the fourth and final trial there will be enough weight on the hanger to allow the glider to accelerate down the track at a sufficient rate. Leaving only the weight of the hanger itself causes the tension in the string to be too slack, and thus will be susceptible to vibration from the air blowing out of the air track, causing a non-uniform acceleration. [Do not attempt to take data with no weights on the hanger.] ANALYSIS OF DATA For each mass combination calculate the applied force, F mh g , and for each run, calculate the value of the measured acceleration. Plot the measured acceleration as a function of the applied force ( F mh g ). You will have at least two measurements of the acceleration for each of the four values of F. You can find the uncertainty for acceleration by propagating the reading errors in the measurements, not by calculating the standard deviation – the standard deviation is not applicable since there are only 2 or 3 trials for each setup. Draw the best-fitting straight line through these points and calculate its slope. Estimate the uncertainty in the slope. (See Fig. 4). You may watch the TA video Exp 3 Uncertainty Video in lab wiki for reference. 131022 5 Brown University Physics Department Physics 0030 Lab 3 9 (x2, y2) 8 7 y2-y1 y 6 5 (x1, y1) 4 x2-x1 3 0 1 2 x 3 4 5 You can approximate the uncertainty in the slope form the maximum and minimum slopes in the range as δm = ( mmax - mmin )/2. Fig. 4 DISCUSSION Equation (3) predicts that your graph is expected to be a straight line, with slope (a / F ) 1 / M passing through the origin. Within the experimental errors: Is your graph linear? Does it pass through the origin? Calculate M from the slope, estimate the uncertainty in this value, and compare with the value of M determined by weighing. Are the two consistent? What are the possible systematic effects which might have influenced your results? Some possible effects which we have ignored are: Remaining friction between track and glider Friction in the pulley Track not quite level Mass of string, which we have ignored String pulling glider possibly not exactly parallel to track. How would these different effects act to systematically change your result? How might you test to see if they are important? 131022 6