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EXPERIMENT 5 Newton’s Second Law Objectives • • Test the validity of Newton’s Second Law Measure the frictional force on a body on a “low-friction” air track Apparatus An air track, one glider, pulley, clamp, masses and two photogate timers will be used (see Figure 1 below). Fig. 1 Theory Newton’s Second Law states that the acceleration of a body is proportional to the net force acting on the body ( a ∝ FNET ) and inversely proportional to the mass of the body 1 ( a ∝ ). Combining these two, we can replace the proportionality with equality. That m is, a= FNET m or FNET = ma FNET is the sum of all of the forces acting on the body. In many textbooks this is denoted by ∑ F . So, Newton’s Second Law is: ∑ F = ma (1) In this experiment a low friction air track will be used to test the validity of Newton’s Second Law. A hanging mass will be attached to a glider placed on the air track by means of a light (negligible mass) string. By varying the mass of the hanging mass we will vary the net force acting on this two body system. We will however, keep the total mass of the system constant. This is accomplished by moving mass from the glider to the hanger. With the air track turned on, the hanging mass will be released and the glider will pass through two photogate timers. The photogate timers will be used to measure two Δx velocities. Recall, v = in our case Δx will be the length of a fin place on top of the Δt glider. If you know the separation between the two photogate timers, you can use an equation from kinematics to determine the acceleration of the glider: v F2 = v I2 + 2 ∗ a ∗ s Where, vF is the velocity measured with the second photogate, vI is the velocity measured with the first photogate, a is the acceleration and s is the distance between the two photogate timers. Solving for the acceleration yields: v F2 − v I2 a= 2s Figure 2: Free body diagrams (2) Separate free body diagrams of the glider and the hanging mass are shown in Figure 2 above. In the figure, f is the net frictional force acting on the body (assume this includes the frictional forces between the airtrack and the glider and the frictional losses in the pulley; N is the upward force the air track exerts on the glider; T is the tension in the string; MGg is the weight of the glider; and MHg is the weight of the hanging mass (FH=MHg). Note: the air track is horizontal and the glider does not accelerate in the vertical direction, therefore N = MGg. Applying Newton’s Second Law to the glider in the horizontal direction and using right as the positive direction yields: +T – f = MG*a (3) If we now apply Newton’s Second Law to the hanging mass, and this time define downward as the positive direction we find: -T + MH*g = MH*a (4) We have no way to directly measure the tension in the string (T), therefore we will combine equations (3) and (4) to eliminate the tension from the resulting equation: FH – f = (MH + MG)*a (5) There are only two unbalanced forces acting on our two-mass system (i.e. the weight of the hanging mass and friction). Notice what equation (5) states: the left hand side is the net force and the right hand side is the product of the system’s mass and its acceleration. This is Newton’s Second Law applied to our two body system. If we rearrange equation (5) we obtain: FH = (MH + MG)*a + f (6) Equation (6) has the same form as the equation for a straight line y = mx + b, where the weight of the hanging mass (FH) plays the role of y and the acceleration (a) plays the role of x! Procedure 1. Set up the air track as shown in Figure 1. With the hanging mass disconnected from the glider and the air supply on, level the air track by carefully adjusting the air track leveling feet. The glider should sit on the track without accelerating in either direction. There may be some small movement due to unequal air flow beneath the glider, but it should not accelerate steadily in either direction. 2. Measure the length (L) of the fin on top of the glider and record it along with its uncertainty in your spreadsheet. See Figure 3. Figure 3 3. Make sure the hook and counter balance are both inserted in the lower hole on the glider. 4. Measure the mass of the empty glider and empty hanger (MG0 and MH0) and record these masses in your spreadsheet. 5. Using 5, 10 and/or 20 gram masses place 40 grams of mass to the glider. Distribute the masses symmetrically so that the glider is balanced. Determine the total mass of the glider (MG0 + the mass you just added) and record this in your spreadsheet. 6. Place 10 grams of mass on the mass hanger. Determine the total mass of the full hanger (MH0 + the mass you just added to the hanger) and record this in your spread sheet. 7. Determine the total mass of your system (MG + MH) – this should be the sum of the masses you entered in steps 5 and 6. Record the system mass in your spreadsheet. 8. Choose a starting position (X0) for the glider near the end of the track. Using the ruler permanently affixed to the air track, measure and record this location in your spreadsheet. Also, measure and record the location of the two photogate timers (X1 and X2) and assign a reasonable uncertainty to these positions (δx). It is very important that your glider always starts from the same location (X0) and that the two photogate timers are not moved. If they are accidentally bumped or moved, return them to their original location. Calculate the magnitude of the displacement between the two timers using S = | X2 – X1 | and record this in your spreadsheet. See Figure 3. In addition, calculate the uncertainty in this displacement δS=2δx. 9. Set your photogate timer to GATE mode and make sure the memory switch is set to on. The GATE mode will only record time when the glider is passing through one of the two photogates. In this mode, the timer will only display the time the glider took to pass through the first photogate (t1). The time the glider took to pass through the second photogate will be added to the memory. Flipping the toggle switch to read will display the total time the glider took to pass through both photogates (tmem). To obtain the time the glider took to pass through the second photogate, simply subtract t1 from the time stored in the photogate’s memory t2 = tmem – t1. The uncertainty in a measurement of time is δt=0.5ms. Using the rules for addition and subtraction, the uncertainty in t2 = 2 δt=1.0ms. 10. With the air supply on, hold the leading edge of the glider stationary at X0; press the reset button on the photogate timer, then release the glider. Make sure the glider does not bounce off the far end of the air track and pass through the second photogate a second time. The time displayed on the photogate’s screen will be the time the glider took to pass through the first photogate (t1). Record t1 in your spreadsheet. Flip the memory toggle switch to read the displayed time (tmem) in your spreadsheet; use the displayed time (tmem) and t1 to calculate t2. 11. Return the glider to X0. Make sure all of the masses are still on the hanging mass hanger and the string is still over the pulley. 12. Move 10 grams from the glider to the hanger. Calculate the total glider mass and the new total hanger mass. (NOTE: the total system mass has not changed.) Repeat steps 10-11. 13. You should have data for 10g, 20g, 30g, 40g and 50g added to your hanger. 14. Have Excel calculate v1, v2 and their respective uncertainties. v= L t and ⎛ δL δt ⎞ + ⎟ t ⎠ ⎝ L δv = v⎜ 15. Have Excel use equation 2 to calculate the acceleration. 16. Have Excel calculate the uncertainty in the acceleration δa. ⎡ δs 2(v 2δv 2 + v1δv1 ) ⎤ + ⎥ v 22 − v12 ⎣s ⎦ δa = a ⎢ 17. Have Excel calculate the weight (FH) of each of the hanging masses. Transfer your data into Kaleidagraph and make a graph of FH vs. a. Make sure you include horizontal error bars associated with the acceleration (a). Questions 1. From the equation of your best fit line and equation 6, what is the net frictional force acting on the glider? What is the significance of this net frictional force? 2. Discuss the consistency of the slope of your best fit line with the theoretical value (see equation 6). 3. Do your results support Newton’s Second Law? Why or why not? . References Instruction Manual and Experiment Guide for the PASCO Scientific Model ME-9206A and ME-9215A Photogate Timers, Pasco Scientific, 1994.