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1 Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force is simply push or pull. Is n object s velocity relted to the pplied force or does the force just chnge the object s velocity? Wht role does mss ply? Isc Newton s 2 nd lw nswers ll of these questions bout the dynmics of n object s motion. Newton s 2 nd lw cn be stted s: F net = m where F net is the net force pplied to n object of mss m nd is the ccelertion of the object. Note this is vector eqution Inclined Plne In the first prt of the experiment, we will mesure the motion nd force on crt s it is pulled up n inclined plne by mss. A schemtic of the configurtion is shown in Figure 4.1. M θ m Figure 4.1: Schemtic of the inclined plne 19
2 To nlyze the sitution, consider the free body digrms of the crt with mss M nd the hnging weight with mss m shown in Figure 4.2. For the hnging mss m in the verticl θ Mg sinθ T θ Mg T mg direction, Newton s 2 nd lw gives: nd Figure 4.2: Free body digrms for M nd m F net = m F net = T mg = m( ) T mg = m (4.1) Where T is the tension in the string, m is the mss, g is the ccelertion of grvity nd is the ccelertion. Note tht the ccelertion is in the negtive direction. For the crt with mss M, consider the forces prllel to the inclined plne. Newton s 2 nd lw gives: nd F net = T Mgsinθ = M T Mgsinθ = M (4.2) where T is the tension in the string which is the sme tension for the hnging mss, is the ccelertion which is the sme for the hnging mss, θ is the ngle the inclined plne mkes with the horizontl direction, nd M is the mss. Eqution 4.1 nd Eqution 4.2 re two simultneous equtions for nd T. Solving for by eliminting T gives: mg Mgsinθ = (4.3) M +m T cn then be found by substituting into Eqution 4.1 or Eqution Atwood Mchine The physicl pprtus for the second prt of the experiment is n Atwood mchine. The Atwood mchine consists of two unequl msses connected by string over pulley. In figure 4.3, m 1 is greter thn m 2 so m 2 moves up nd m 1 moves down. To understnd the Atwood mchine using Newton s 2 nd lw, consider the motion of the two msses moving under the influence of grvity. The string connecting the msses hs tension, T, which in both cses is directed upwrd. Both msses hve n ccelertion of equl mgnitude but m 2 is ccelerting upwrds nd m 1 is ccelerting downwrd. Free 20
3 m 1 m 2 Figure 4.3: Bsic schemtic of the Atwood mchine m 2 T m 1 T F = 2 g m 2 F 1 = g m 1 Figure 4.4: Free body digrms for m 1 nd m 2 body digrms which show the forces cting on the individul msses nd their ccelertions re shown in Figure 4.4. For m 1, Newton s 2 nd lw gives: Applying Newton s 2 nd lw to m 2 yields: T gm 1 = m 1 (4.4) T gm 2 = m 2 (4.5) Note tht the signs for the ccelertion, tension, nd grvittionl force hve been explicitly included in equtions eqution 4.4 nd eqution 4.5. These two simultneous equtions cn be solved for the ccelertion nd tension. The ccelertion,, is given by: = (m 1 m 2 )g (m 1 +m 2 ) (4.6) 4.3 Procedure First we will mesure the ccelertion nd tension in the inclined plne configurtion nd then mesure the ccelertion of n Atwood mchine Inclined Plne The dynmic crt system is shown in Figure 4.5. The pulley t the end of the trck is soclled smrt pulley. The smrt pulley consists of smll plstic wheel nd photogte. 21
4 The pulley wheel is designed to hve smll mss nd low friction in the berings. The photogte hs n infrred bem on one side which is detected on the opposite side. The spokes of the wheel llow the bem to pss through the open region between the spokes or block the bem. The motion of the wheel cn then be detected s the mss flls. The force sensor is ttched to the crt. A string is ttched to the force sensor which is positioned over the smrt pulley nd is ttched to the flling mss. The end of the trck with the smrt pulley is elevted by wood block. Figure 4.5: Apprtus for the inclined plne experiment Remove the wood block from under the end of the trck with the smrt pulley. Verify the trck is level. The crt should not move either direction if the trck is level. The feet of the trck cn be djusted if necessry to level the trck. Mesure the distnce between the feet, d, of the trck. Mesure the thickness of the wood block, t. Record both vlues below. Distnce between feet of the trck (d) Thickness of wood block (t) The elevtion ngle of the trck, θ, is sin 1 (t/d). Clculte θ nd record it below. Elevtion ngle (θ) Weigh the glider nd ttched force sensor. Record the vlue below. Mss of crt nd force sensor (M) Open the file 2nd lw in Cpstone. A grph of force vs time nd velocity vs time will pper. Plce the block under the feet of the trck nerest the smrt pulley. Plce the crt on the trck. Without the string ttched, press the Tre button on the force sensor. Attch the string to the force sensor, over the smrt pulley nd to the mss hnger with 0.1 kg of mss ( 50 grms mss nd 50 grm hnger). Hold the crt t the end of the trck wy from the smrt pulley with your hnd. The string must be on the pulley of the smrt pulley. 22
5 Click record nd then relese the crt. Click stop fter the crt reches the end of the trck. Identify nd highlight the region on the velocity vs time grph where the crt ws moving. Fit this region with liner fit. The slope of the line (m) is the ccelertion (). Identify nd highlight the region of the force vs time grph where the crt ws moving. The sttistics function men ( ) for the force during this time is the tension in the string (T). Record the vlues s tril 1 in the dt tble. (Note tht the force vlue is negtive becuse of the orienttion of the force sensor. In the coordinte system defined in the introduction, T is positive.) Repet the mesurement two more times for totl of 3 mesurements. Record the dt in the dt tble. Clculte nd record the men of the ccelertion nd tension. Tril Accelertion () Tension (T) Sum Tble 4.1: Dt tble for the inclined plne dt Men of ccelertion () Men of Tension (T) Clculte the ccelertion from Eqution 4.3 nd find the tension using either Eqution 4.1 or Eqution 4.2. Record the vlues below. Clculted ccelertion Clculted Tension Find the percent error between the clculted nd experimentl vlues (men) for nd T Atwood Mchine % error in %error in T The two msses of.200 kg nd.205 kg re hung over the smrt pulley which detect the motion of the string s the msses move (See figure
6 Figure 4.6: The Atwood mchine. The smrt pulley is mounted t the top. The two msses connect by string over the smrt pulley The two msses hnging from the string re kg (brss) nd kg (stinless steel). Do not weigh the msses since they re difficult to replce on the string. Record the vlues of the msses. Mss of m 1 Mss of m 2 Open the file twood.cp in Cpstone. Two plots will pper on the screen. One plot is velocity vs time. The other plot is position vs. time. Hold the smller brss mss ner the tbletop. It should be positioned so it does not swing when relesed. Relese the mss from rest nd click the record button. Click stop just before the lrger mss hits the tble. Highlight the time region of interest on the position vs. time plot. Fit this region with qudrtic fit. The qudrtic term in the fit ( A ) is 1 2. Accelertion from qudrtic fit of position vs time grph Highlight the time region of interest on the velocity vs time plot. Fit the region of interest with liner fit. The slope of this line is the ccelertion,, of the msses. Record this vlue of the ccelertion in the dt tble below s tril 1. Men of 5 trils Repet the mesurement 4 more times for five totl mesurements. Using the vlue of the ccelertion from the five fits to the velocity vs time grph, determine the men of your five mesurements. 24
7 Tril Accelertion Sum Clculte the percentge error between the men vlue of the ccelertion from the dt tble nd the theoreticl vlue clculted from Eqution 4.6. Accelertion from Eqution 4.6 Percentge error Sve one exmple set of dt (.cp file) from the five mesurements nd submit them to your TA by emil ttchment. (Be sure to lbel the the file with your nme, your lb prtner s nme, nd the lb nme.) Questions 1. Solve eqution 4.1 nd eqution 4.2 for the ccelertion of the crt (eqution 4.3) nd the tension in the string. 2. Solve eqution 4.4 nd eqution 4.5 for the ccelertion of the msses (eqution 4.6) nd the tension in the string. 3. The mss of the pulley nd the friction in the berings of the pulley nd crt re smll. Is your dt consistent with the pulley or crt hving smll but nonzero effect on the motion? 25
8 4.4 Conclusion Write detiled conclusion bout wht you hve lerned. Include ll relevnt numbers you hve mesured with errors. Sources of error should lso be included. 26
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