NEWTON'S SECOND LAW (with friction) OBJECTIVE: To examine our model of friction by investigating the sliding of a block up a ramp. THEORY: The application of Newton's 2nd Law to this block-ramp system yields similar results as in last week's set-up except that in this lab we are including friction. Mgsine Mgcose FIGURE 1 Consider Figure 1 where the block is moving up the ramp. A hanging weight (mass mz) is attached to the string. The string exerts a tension (7) on both objects. Application of Newton's 2nd Law gives for the block EFX = T - fx - Mg sin0 = Ma (1) EFy = N - Mg cose = 0 (2) and for the hanging weight: EFX = 0 (3) EFy = -T + mag = mza (4) (Note that we have taken down to be the positive direction for the hanging weight so that the accelerations of the block and hanging weight have the same sign.) Eq. (2) yields the normal force as: N = Mg cos0 (5) Solving for the kinetic friction force by combining Eqs. (1)& (4) gives: fx = (m2 - M sin0)g - (mz + M)a (6) For each ramp angle, then, there is a different sized normal force and kinetic friction force as described by Eqs. (5)&(6). However, our model of friction states that the ratio of the kinetic friction force to the normal force (which is the coefficient of kinetic friction) is constant, LLK = f* / N . (7 ) The coefficient depends only on the kind of surfaces that are in contact. It is independent of contact area. There is a slight dependence of ul on speed, with Lu decreasing as the object travels faster, but we will ignore this effect since the block will not be moving at high speeds.Newton's 2" Law (with friction) 2 PROCEDURE: 1. (a) Weigh your block to determine its mass, M. (b) Raise the ramp to 25. (c) Place the block on the ramp with its largest area touching the ramp and the string tied to the eyelet so that the string is parallel to the ramp when placed over the pulley. (d) Hang a weight, mz, on the string so that the block slides up the ramp a measured distance s. (The yellow marks on the ramp are 80 cm apart.) Measure the time, t, for this motion. Select a weight so that (> 1 s. (e)Estimate the uncertainty in the time, ot. (f) Record the value of mz. 1-1. Calculate the acceleration of the block using your values of s and t and the equation of motion for constant acceleration, a = 2s / f. 1-2. Calculate the uncertainty of the acceleration, da . From the previous equation for the acceleration, this uncertainty is da = 2 a ( ot /t) 1-3 Calculate the normal force in Newtons using Eq. (5) 1-4. Calculate the kinetic friction force in Newtons using Eq. (6). 1-5 Calculate the uncertainty in the friction force, ofk. From Eq. (6), this uncertainty is & fx = (m2 + M) da 1- 6 . Calculate the coefficient of kinetic friction using Eq. (7). 1-7. Calculate the uncertainty of the coefficient, o jk. From Eq. (7), this uncertainty is SULK = Ofk / N. Express your value of the coefficient as Jk 1 8 Ilk. 2. We now see if the area of contact between the ramp and block makes a difference. (a) Repeat Step 1, but this time place the block on its side. Attach the string to the other eyelet so that the string remains parallel to the ramp. Use the same hanging weight. 1-1. Are the two coefficient of friction values the same when their respective uncertainties are considered? 2-2. Is the observation made in 2-1 consistent with our model of friction? 3. We now see how a change in mass affects the results. (a) Reattach the string to the original eyelet and place the block on the ramp in its original orientation. (b) Add 200 gm of mass to the top of the block. Note that M is now the block's original mass plus the 200 gm. (c) Repeat Step 1. You may have to increase the hanging mass. 2-2. Is the normal force larger, smaller, or the same compared to when the block was lighter? Does this agree with theory? 2-2. Is the kinetic friction force larger, smaller, or the same? Does this agree with theory? 3-3. Is the coefficient of kinetic friction larger, smaller, or the same? Consider the uncertainties of each value. Does this agree with our theoretical model? 4. We now see how the ramp angle affects the results. (a) Remove the extra 200 gm from the block and measure the times for the block to travel the distance s for ramp angles of 35, 45, and 509. You may have to adjust the hanging weight. 1-1. Calculate the acceleration and friction force values for each ramp angle. 4-2. Calculate the normal force values for each ramp angle. Is the normal force constant? Does it increase or decrease as the ramp angle increases? 4-3. Calculate the coefficients of friction and the uncertainties in the coefficients as you did in Step 1. 4-4. Plot the calculated fx VS. / values. Include the other data points from Steps 1&3.Newton's 2"" Law (with friction) 3 Also; include the point (0,0). Calculate the slope of the line. According to our theoretical model. the slope is the coefcient of kinetic friction Since fr: = .W N. Record the value of Lu: obtained from your line. 4-5. The inclusion of the point (0.0) on your plot will help you in determining the best line. It is a valid point since if there is no normal force. there cannot be any tn'ction. ft is necessary to add it because it is not practical to obtain normal force values for this experimental setup that are less than approximately 1.3 Newtons. Why not? (Hint: 596 Bi (5)) REPORT: 1. There is a large amount of raw data and calculated data in this experiment. Be sure that you organize it clearly in your report. Atable or tables would be best for displaying your results 2. Answer a_ll of the questions posed in the procedure part. DO NOTSIMPLYNUMBER THE QUESTIONS AND WRITE DOWN ANSWERS. WRITE COMPLETE SENTENCES 1N YOUR DISCUSSION THATFULLYADDRESS AND ANSWER THE QUESTIONS. 3. What can you conclude about our model for friction? Review your results carefully before answering this question. Ask yoursetf ifany discrepancies found can be fully explained by experimental uncertainties or by assumptions made in the model. Title: Objective: To examine our model of friction by investigating the sliding of a block up a ramp. Raw Data: Date: Distance of travel 5 : 0.8 m Ramp Block Hanging Times Name: Angle 8 Height h Mass M Mass m; 11. [2,13 Run (deg) (cm) (kgl (kg) (5) 1.63,1.53,1.55 Fanny's). 'l 25 42.3 0.2338 0.2 2 25 42.3 0.2338 0.2 1'73' 1'66' 1'50 3 25 42.3 0.4338 0.35 1'70' 1'68' 1'66 4 35 57.4 0.2338 0.22 1'82' 1'87' 1'86 5 45 70.7 0.2338 0.25 1'66' 1'75' 1'71 Sample: h = (100 cm)iV= (100 cm) sin (25) = 42.3 cm Calculated Data: Ave. Time Acceleration Acceleration Normal Force t a Uncertainty N Sample Calculations: Run (s) (m/s^2) (N) (m/s^2) Average Time 1.57 0.65 0.08 2.08 t = (t1 + t2 + 13) / 3 = (1.63s + 1.53s + 1.55s) / 3 = 1.57s 2.08 2 1.63 0.60 0.07 Acceleration 0.07 3.85 3 1.68 0.57 a = 2s/ t2 = (2 * 0.8m)/(1.57s^2) = 0.65 m/s^2 0.47 1.88 4 1.85 0.05 Acceleration Uncertainty 5 1.71 0.55 0.06 1.62 da = 2a (8t / t) = (2 *0.65m/s^2)(0.1s/1.57s) = 0.08m/s^2 Normal Force N = Mg cos 0 = 0.2338kg * 9.8 * cos(25degrees) = 2.08N Friction Force Friction Force Coefficient Friction Force Ik Uncertainty Run (N) ofk (N) fx = (m2 - M sin 0)g - (m2 + M)a = (0.2kg - 0.2338 * sin(25deg)9.8- (0.2kg+0.2338)0.65m/s^2 = 0.710N 1 0.710 0.035 0.341 0.017 Friction Force Uncertainty 2 0.731 0.030 0.35110.014 of* = (m2 + M) da = (0.2kg + 0.2338kg)(0.08m/s^2) = 0.035N 3 1.187 0.055 0.30810.014 Coefficient of Friction 4 0.629 0.023 0.33510.012 HK = fx/ N = (0.710N)_(2.08N)= 0.341 uk 5 0.564 0.029 0.34810.018 Coefficient Uncertainty Suk = ofk / N = (0.035N/2.08N) = 0.017 8ukPlot: Plot 1: (Plot the calculated (jars. N values. Also, include the point (0,0). Calculate the slope of the line. According to our theoretical model, the slope is the coefcient of kinetic friction since 3': = pk N. Record the value of pk obtained from your line.) Results: Plot 1 : (Report the slope of the line which is the coefficient of friction.) Run 2: (We now see if the area of contact between the ramp and block makes a difference. Are the two coefficient of friction values the same when their respective uncertainties are considered? Is the observation made in 2-) consistent with our model of friction?) Run 3: (We now see how a change in mass affects the results. Is the normal force larger, smaller or the same compared to when the block was lighter? Does this agree with theory? Is the kinetic friction force larger, smaller, or the same? Does this agree with theory? Is the coefcient of kinetic friction larger, smaller, or the same? Consider the uncertainties of each value. Does this agree with our theoretical model?) Runs 4-6: (We now see how the ramp angle affects the results. Is the normal force constant? Does it increase or decrease as the ramp angle increases? What happens to the friction force as the ramp angle increases? What should happen to the coefficient of friction as the ramp angle increases? Do you observe this trend if you consider the ranges of each coefficient?) Uncertainties: (What were the sources of measurement uncertainty? Which source was the largest contributor to measurement uncertainty? Does the measurement uncertainty alone account for any differences between calculated values and accepted values?)