Two blocks connected by a string with variable tension: A dynamic case
aa r X i v : . [ phy s i c s . e d - ph ] A ug Two blocks connected by a string with variabletension: A dynamic case.
H. J. Herrera-Su´arez ,a & M. Machado-Higuera ,b & J. H.Mu˜noz ,c Universidad de Ibagu´e, Facultad de Ciencias Naturales y Matem´aticas,Ibagu´e-Colombia, Carrera 22 Calle 67, barrio Ambal´a Universidad del Tolima, Departamento de F´ısica, Ibagu´e-Colombia, Barrio SantaHelena Parte Alta.E-mail: a [email protected] & b [email protected] & c [email protected] Abstract.
In this paper, the dynamic case of a system made up of two blocksconnected by a string over a smooth pulley is revisited. One mass lies on a horizontalsurface without friction, and the other mass has a vertical displacement. The motionequation is obtained and its solution is determined using the
Mathematica package.Also an experimental montage for this system is made and experimental data for thevertical position y in function of the time t are obtained using a Data AcquisitionSystem and the Tracker video analysis. The relation y vs t can be represented by apolynomial of degree six. An average relative error of 3 . . Mathematica and the data taken from the
Tracker (Data Acquisition System).
Keywords : newtonian mechanics; tracker; data acquisition system.
1. Introduction
In this work, a system of two blocks of masses m and m connected by a string overa smooth pulley and subjected to variable acceleration is studied. The mass m issuspended, and the mass m moves on a horizontal surface without a coefficient ofdynamic friction. The string is extensionless, and uniform, and its mass is negligible.Figure 1 shows the forces acting on this system.Unlike other systems made up of two masses tied to a rope that passes through apulley (such as those shown in Figure 2), this problem is significant because the tension T , the normal force N and the acceleration are not constant because they depend on thevariation of the angle θ . This system is a more complex progression of the four commonconfigurations shown in Figure 2.The static situation of this problem has been studied previously in somefundamental physics textbooks [1–6], in several papers [7–12] and on the website ofA. Franco [13]. The dynamic case, in which there is no friction between the horizontal wo blocks connected by a string with variable tension: A dynamic case. Figure 1:
Two blocks tied to an extensionless string.
Figure 2:
Systems made up of two masses tied to a rope that passes through a pulley.The tension and the acceleration are constant.surface and the mass m , was proposed as a problem in Serway-Jewett’s physics textbook[1] and its solution appears in the Instructor’s Solution Manual of the same author [2].In this paper, this case is revisited with the purpose of performing a complete analysisto this system. The motion equation is explicitly obtained, and its solution is foundusing the Mathematica package. An experimental montage is made, and experimentaldata for the position in function of the time for the mass m are got using a DataAcquisition System (DAS) and the Tracker video analysis. The experimental resultsare then compared with the theoretical solution.This paper is organized as follows. In Section 2, the experimental arrangementbuilt by the authors to represent the system shown in Figure 1 is displayed; in Section3, the theoretical analysis is presented, and the results are briefly discussed in Section4. Finally, in Section 5, some concluding remarks are given. wo blocks connected by a string with variable tension: A dynamic case.
2. Experimental arrangement
Figure 3 shows the experimental arrangement used to represent the system in Figure1. The two masses are m = 574 .
12 g and m = 100 g (measured with a WTC 2000precision balance [20]), and the length of the string is l = r + y = 2 .
09 m and h = 0 .
98 m.The mass m moves down in the vertical direction, and the mass m moves horizontallyover the linear air track, reference 11202 −
88, of the company Phywe [14]. The DASused is made up of: a sensor-CASSY 2, reference 524013 [15]; a timer S [16]; a multicore cable, 6 pole, 1 . m and the air track. The data for the variable y in function of the time wereregistered for the DAS with the initial condition y (0) = − .
255 m. The movement ofthe two masses was also recorded with a smartphone, and the video was analized usingthe
Tracker video analysis [21].
Figure 3:
Experimental arrangement: (a) mass m ; (b) mass m ; (c) combinationspoked wheel; (d) air track; (e) sensor CASSY 2; (f) timer S; (g) multi core cable, 6pole, 1.5 m; (h) computer; (i) air supply, and (j) combination light barrier.
3. Theoretical analysis
The system displayed in Figure 1 is analyzed using Newton’s second law. The sum ofthe forces for the particle m in the x direction is wo blocks connected by a string with variable tension: A dynamic case. T cos θ = m a x ; (1)and the sum of the forces for the mass m in the y direction gives T − m g = − m a y . (2)From this equation, the tension T is obtained and by substituting it in Equation(1) the following is got( m g − m a y ) x √ x + h = m a x , (3)where cos θ = x/r = x/ √ x + h (according to Figure 1).Considering Figure 1, the length l of the rope is given by l = √ x + h + y. (4)From this equation the following is obtained x = p ( l − y ) − h . (5)On the other hand, if differentiated with respect to time in Equation (4), the equationbelow is got v y = − xv x ( x + h ) − , (6)where v x = dxdt and v y = dydt are the speeds of mass m and m , respectively.Now, by differentiating this expression with respect to time, after replacing x and v x , by means of the Equations (5) and (6), respectively, the below expression is got a x = − h v y (cid:0) ( l − y ) − h (cid:1) + ( l − y ) a y (cid:0) ( l − y ) − h (cid:1) , (7)where a x = dv x dt and a y = dv y dt are the accelerations of mass m and m , respectively.If v y = 0, then a x = − ua y with u = ( l − y )(( l − y ) − h ) − . This result agrees with theone presented in the references [1, 2].By Replacing the last expression for a x in Equation (3), the following expression isobtained: a y = m g (cid:0) ( l − y ) − h (cid:1) + ( l − y ) m h v y (cid:0) ( l − y ) − h (cid:1) (cid:0) m (cid:0) ( l − y ) − h (cid:1) − m ( l − y ) (cid:1) . (8)This differential equation gives the equation motion for y . wo blocks connected by a string with variable tension: A dynamic case.
4. Results and discussion
The evolution of the variable y in function of the time t has been obtained via threedifferent ways: (i) employing the Tracker , (ii) manipulating the DAS, and (iii) using the
Mathematica package.As mentioned in Section 2, the movement of the masses m and m were recordedwith a smartphone and the video was analyzed with the Tracker . Figure 4 (see left side)shows a picture of the movement of the mass m using this computational tool. Thevertical displacement of the particle m in function of the time, y vs t , is displayed inthe top-right of this figure, and the data are shown in the bottom-right. Figure 5 showsthe graph of y vs t obtained with the Tracker (see the solid black circles). On the otherhand, the software CASSY was also used to obtain y vs t from the DAS (see the solidblue circles). Figure 4:
An image of the movement of the mass m using the Tracker .The solution of the differential equation shown in Equation (8), was obtained bymeans of the
Mathematica , using the commands NDSolve and Plot[Evaluate[y[t] /. s,t, 0, 1]. This solution is displayed in Figure 5 (see the solid red circles).Figure 5 was obtained using the OriginPro package [22]. This software allows theresults obtained by the
Tracker , the DAS, and
Mathematica to be presented in a singleimage. The experimental data confirm that the motion of the mass m is neither uniformnor uniformly accelerated. wo blocks connected by a string with variable tension: A dynamic case. Mathematica DAS Tracker y ( m ) Time (s)
Figure 5:
Graph of y vs t for the mass m . The solid black (blue) circles representthe experimental data obtained with the Tracker (DAS). The red circles represent thesolution of the differential equation obtained with
Mathematica .The obtained data with the DAS and the Tracker and the acquired result withMathematica, were fitted by means of the OriginPro. The best fit is given by apolynomial of degree six with a correlation coefficient of, approximately, 0.999. Thecorresponding expressions are display in the Table 1. The curves and the errors shownin Figure 6 were obtained from these equations. The results obtained with
Mathematica and the
Tracker (DAS) were compared (see Figure 6a (Figure 6b)), giving that theaverage relative error of the theoretical results acquired with
Mathematica in relationto the data taken with the
Tracker and the DAS is 3 .
61 % and 10 .
14 %, respectively.It is important to note that both measured curves indicate lower accelerations thanthe calculated indicating that there is some small friction. Further, the accelerationobtained with the DAS is slower than the one obtained with the Tracker probablybecause of cord slippage around the pulley. The curves show good agreement at thebeginning and then diverge because the increasing angle is altering whatever frictionthere is. wo blocks connected by a string with variable tension: A dynamic case. Mathermatica Tracker y ( m ) Time (s) (a)
Mathermatica DAS y ( m ) Time (s) (b)
Figure 6:
Comparison of the obtained results from Mathematica and the Tracker (a)and the DAS (b).
Table 1: y vs t for the obtained data with the Tracker, the DAS and the theoreticalsolution given by Mathematica. Mathematica y = − . × − t + 5 . × − t − . × − t − . t +2 . × − t − . y = 1 . × − t − . t + 0 . t − . t − . t + 0 . t − . Tracker y = − . × − t + 0 . t − . t +0 . t − . t − . t − .
5. Concluding remarks
The authors studied the dynamic situation of a block of mass m on a horizontal planebeing pulled at an angle θ with the horizontal by a tension due to a suspended mass m ,without considering the friction between m and the horizontal plane. The experimentalresults were obtained using the Tracker and the DAS shown in Figure 3, and thetheoretical solution to the motion equation was found using
Mathematica .The movement of the mass m along the y-axis is neither uniform nor uniformlyaccelerated. The best fit for y in function of the time is a polynomial of degree six. Thetheoretical prediction obtained with Mathematica gives a better agreement with datataken using the
Tracker than using the DAS.This work is worthwhile because it allows the articulation between experiment andtheory, facilitating students’ understanding of the physics behind the theory. Besides,it allows one to check the experimental viability of theoretical problems proposed inphysics textbooks. wo blocks connected by a string with variable tension: A dynamic case. References [1] Serway R A and Jewett J W 2014
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