acceleration of light pulley is

Adjust the initial position of the box, the mass of the box, and the spring constant. Two blocks of mass m1 and m2 attached by mass less string ... computer continuously monitored each photogate's light beam. Override with M204. There is no slipping between the rope and the pulley surface. 5 since the weight of this mass is equivalent to the frictional force of the pulley. Adding a a m s T N Block T f F m a T a Block m g T m a T a f N m g N distance 2.6 m apart. The particles are attached to the ends of a light inextensible spring. Experiment 1b: Light Gates Measuring Displacement and Velocity Variables Independent Variable: The force applied to the object. Control Variables: The mass of the object. Share 0. Draw free-body diagrams for each of the two blocks and the pulley. A trolley experiences an acceleration when an external force is applied to it. This physics video tutorial explains how to calculate the acceleration of a pulley system with two masses with and without kinetic friction. Determine the acceleration of the system. Since the string is inextensible, therefore both the masses will have same acceleration a. Two blocks are connected by a very light string passing over a massless and frictionless pulley (Figure ).The 20.0-N block moves 75.0 cm to the right and the 12.0-N block moves 75.0 cm downward.A. We would release the weight from a known height and measure the time for it to reach the ground to calculate acceleration. Block B, of mass mB, hangs from a light string that runs over a pulley and attaches to block A, as shown above. The system is held at rest with the string taut, the hanging parts of the string vertical and with A and B at the same height above a horizontal plane, as shown in Figure 4. Find acceleration of the cart. What is the acceleration of the masses when left f asked May 9, 2019 in Physics by AarohiBasu ( 85.1k points) It is calculated using the following formula: acceleration = change in speed ÷ time taken. A counterweight of mass mis attached to a light cord which is wound around a pulley as shown in the gure below. MMS-I , a = 3.7632 ms −2, T =18.1104 N A θ P B The acceleration of gravity is 9.8 m/s2. There are two ways to go about solving it. $\begingroup$ If the pulley is "rolling up" the left rope, like a yo-yo, then it is not an ideal pulley and the two tensions cannot be the same. A trolley experiences an acceleration when an external force is applied to it. [4 points] In both cases shown, a pulley is suspended by a rope from a During movement planning, Marlin constrains the default accelerations to the maximum acceleration of all axes involved in the move. If the masses accelerated at Acceleration is the rate at which velocity changes. Pulley E … In case I a force of 50 newtons is applied to the cord. Textbook solution for University Physics (14th Edition) 14th Edition Hugh D. Young Chapter 5 Problem 5.114CP. negligible mass passing over a pulley of Radius r and moment of inertia . Step 3: Analyze the pulley. plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in the figure above. Three box system problem. Two blocks are connected by a very light string passing over a massless and frictionless pulley (Figure ). Transcript. → Freebody diagrams are profoundly helpful in visualizing the many forces at play here. a) 0.2 m/s2 b) 9.8 m/s2 c) 5 m/s2 d) 4.8 m/s2 2. 1.25, so that the tension (T) in both the strings is same. A uniform chain of mass m hangs from a light pulley, with unequal lengths of the chain hanging from the two sides of the pulley. The combined mass of the stand and the pulley is .A light string passes over the smooth pulley and two masses m and M (gt m) are connected to its ends (see figure). `8:1` B. Created by David SantoPietro. The moment of inertia of the two wheels together is I CM = 40 … ANSWER: Exercise 10.16 Description: A 12.0kg box resting on a horizontal, frictionless surface is attached to a 5.00kg weight by a thin, light wire that passes without slippage over a frictionless pulley (the figure ). The friction between the mass and the surface is represented by a friction coefficient mu= . (Note: The mass of a solid disc pulley is approximately 100 g.) (If you are using a red, spoked pulley, m equiv is approximately 30 g.) Determining the acceleration with a stopwatch and meter stick The acceleration may be determined by measuring the time, t, required for the mass to fall from rest a specific distance 'y. A light, inextensible string passes over a smooth light pulley. The name was later revived in the Type 121 Europa S, a sports car based on the Lotus Elise produced from 2006 to 2010. Calculus – a strand of mathematics that calculates rates of change and helps find lengths, volumes and areas. Lubricate the pulley to reduce the force of friction which acts against the weight added. The force exerted by the moving chain on the pulley is A) mg B) > mg C) < mg D) either (b) or (C) depending on the acceleration of the chain. A mass is connected by a light string that passes over a pulley of mass M to a mass sliding on a frictionless incline as shown in the figure. (1) T--3g = 3a. (i) When unequal masses m 1 and m 2 are suspended from a pulley (m 1 > m 2) m 1 g – T = m 1 a, and T – m 2 g = m 2 a On solving equations, we get (ii) When a body of mass m 2 is placed on a frictionless horizontal surface, then Mass Pulley System acceleration, a = The elevator is going up with an acceleration g/10, the pulley and the string are light and the pulley is smooth. If the acceleration of the system is `g//8`, then the ratio of the masses is A. Sketch a free-body diagram for the pulley. Consider an atwood machine (this same system) where the pulley is not accelerating. = v R. A trolley experiences an acceleration when an external force is applied to it. Acceleration – the rate an object changes its speed at. The cart was released from rest near one of the photogates. We have step-by-step solutions for your textbooks written by Bartleby experts! At each end of the string there is a particle, as in Worked Example 1. Pulley E is fixed but pulleys C and D are free to move. We also weighed the keys on a scale, so I know they actually weigh 37g. (mass3kg will go up with accln. Taking the coe cient of kinetic friction as 0.185, nd the tension in the string. acceleration. The weight of the hanging mass provides tension in the string, which helps to accelerate the cart along the track. Answer (1 of 3): Note: The forces m_1g and m_2g act at the centre of the respective masses while the force F acts at the centre of the pulley although the figures show them to be acting on the surface. A light string is attached to the cart and passes over a pulley at the end of the track and a second mass is attached to the end of this string. The pulley is a … Apparatus and Materials. (b) Tension of the cord. Numerical example: M = 210 g and m = 200 g. a g * 10 410 0.24 m/s2if mp= 0. a g * 10 510 0.19 m/s2if mp= 200 g. Determine (a) the tension in the string, (b) the acceleration of each object, and (c) the distance each object will move in the first second of motion if they start Gfrom rest. Set the software to record data. Acceleration in an Atwood’s machine II. `9:7` C. `4:3` D. `5:3` The high-efficiency 3-axis gear train and the optimized motor have realized compact size, light weight and high efficiency, and can be mounted on the front or rear of a vehicle. Take … Answer. Problem # 1 A block of mass m is pulled, via pulley, at constant velocity along a surface inclined at angle θ. The pulley rotates about a frictionless axle and has a moment of inertia of 0.300 k∙m2 and a radius of 0.500 m. Assuming . a) Find the magnitude of the acceleration of the two masses if the coefficient of kinetic friction between the inclined plane and mass m 1 is equal to 0.4. b) Find the magnitude of the tension in the string. The coefficient of kinetic friction between m2 and the incline is 0.1. Treating the pulley as a uniform disk, find the downward acceleration of the weight and the tension in the cable. $\begingroup$ If the pulley is "rolling up" the left rope, like a yo-yo, then it is not an ideal pulley and the two tensions cannot be the same. In this video David explains how to find the acceleration of two masses hanging from a pulley (using the easy method). Pull the trolley back so that the mass is raised to just below the pulley. The pulley is a thin hoop of radius Rand mass M. The spokes have negligible mass. Determine: A) Tension in the rope B) Acceleration with which the body moves up the plane and C) The distance moved by the body in 3 sec after starting from rest. The 2.8 kg block is the lower one. T1-m1g=m1a, The acceleration of the $2 \ kg$ block if the free end of string is pulled with a force of $20 \ N$ as shown is computer continuously monitored each photogate's light beam. Particle B has a mass of 11 kg and particle C has a mass of x kg. Because velocity is a vector quantity, acceleration is also a vector quantity. A 9.80-kg hanging object is connected by a light, inextensible cord over a light, frictionless pulley to a 5.00-kg block that is sliding on a at table. Hint: Think carefully about the acceleration constraint. (the pulley is ! Dear student, ... let us assume the acceleration of pulley B be c (downwards). This problem explores how the relative masses of two blocks affect the acceleration of the blocks. Most people choose (+) to be the direction that the system will accelerate. Using the acceleration, we are instructed to find the mass of the unknown weight. You have a cart with mass M sitting on the table with a string attached to it. Let mass m1 is greater than mass m 2 . When the block A is sliding on the table, the tension in the string is: David shows the easier way to find the acceleration of two masses connected by a rope. From these equations, you can determine the acceleration of the masses and … Consider this situation. In the system below, blocks of masses m 1 = 10 Kg and m 2 = 30 Kg are linked by a massless string through a frictionless pulley. Release the trolley. In the example, therefore, down on the 10 kg side is (+) and down on the 5 kg side is (–). Remember that if two objects hang from a massless rope (or string, cable etc.) In the system below, blocks of masses m 1 = 10 Kg and m 2 = 30 Kg are linked by a massless string through a frictionless pulley. Treating systems (the hard way) Treating systems (the easy way) This is the currently selected item. The objects are released from rest, and m2 slides 1.00m down the incline in 4.00s. On this page I put together a collection of pulley problems to help you understand pulley systems better. (a) Find the position of the particle at t = 2.60 s. A 9.00 kg mass is connected by a light cord to a 1.00 kg mass on a smooth surface as shown in the figure. In print/travel moves, DEFAULT_ACCELERATION and DEFAULT_TRAVEL_ACCELERATION apply to the XYZ axes. The acceleration of light pulley is A mF B 2mF C 4mF D 8mF Hard Open in App Solution Verified by Toppr Correct option is C 4mF If we take acceleration … It was stopped just before it reached the pulley. Two masses m = 5 kg and m2 = 4.8 kg tied to a string are hanging over a light frictionless pulley. [Hint: First determine the acceleration of the lighter mass and then the velocity at the moment the heavier hits the ground.] It is attached by a rope over a pulley to a mass of kg which hangs vertically. In retraction moves, DEFAULT_RETRACT_ACCELERATION applies only to the E-axis. The The pulley has a diameter d=2. The system is released from rest. (m 1 =0.15 kg, m 2 =0.10 kg, m 3 =0.10 kg, r=0.10 kg, g=10 m/s2) Free body diagrams of two masses and the reel: and compute the predicted acceleration from Newton's second law (Eq. Let us understand that for a stationary pulley when m_1 and m_2 move, they will move in … –A mass (M 1 = 6.0 kg) is connected by a light cord to a mass (M 2 = 4.0 kg) which slides on a smooth surface, as shown in the figure. Find the reading of the spring balance shown in the following figure. Thus the total force on each block will be zero. The lighter block will have the Positive Acceleration that is +2.45 m/s square. To calculate the acceleration of the masses, we note that the masses will accelerate at the same rate together. One block, of mass , rests on a smooth horizontal surface and is attached to a cord of negligible mass with tension . side pulley (see Fig.8.3). Whenever a metal strip ("flag") attached to the glider passed through a light beam, Assume that the cable does not slip with respect to the pulley. What is the acceleration of block A as it slides on the frictionless table? NeuLog data acquisition and analysis application for MACs. 5 since the weight of this mass is equivalent to the frictional force of the pulley. Choose a positive direction, and apply Newton’s Second Law for rotation. The acceleration of M 2 is 3.5 m/s2. (Note: The mass of a solid disc pulley is approximately 100 g.) (If you are using a red, spoked pulley, m equiv is approximately 30 g.) Determining the acceleration with a stopwatch and meter stick The acceleration may be determined by measuring the time, t, required for the mass to fall from rest a specific distance 'y. A 1.4 kg block and a 2.7 kg block are attached to opposite ends of a light rope. Note that both masses feel the same tension because it is undiminished throughout the light string. This is the currently selected item. First of all, a pulley is a wheel on an axle or shaft designed to support movement and change direction of a belt or taut cable. After release, determine the acceleration of the system and the tension in the string. String . Apparatus and Materials. Setting the two equations equal to each other, Mg−Ma=mg+ma⇒(M−m)g=(M+m)a. Now that the acceleration has been found from Equation 5, its value can be substituted into Equation 4 in order to determine the tension. Taking the coe cient of kinetic friction as 0.185, nd the tension in the string. If the total resistance force to the motion of the cart is 0.72 N, then what is the cart's acceleration? The angle θof the incline is 20º. The cart was released from rest near one of the photogates. Two bodies, m1= 1kg and m2=2kg are connected over a massless pulley. What is the acceleration of the masses when they are free to move (g = 9,8 m.s?) connected by a light inextensible string which passes over a smooth light fixed pulley. a) what is the acceleration of each mass? 5). A system comprising blocks, a light frictionless pulley, a frictionless incline, and connecting ropes is shown in the figure. As it was accelerating it passed through both photogate beams. T. A. equal to the tension in string . phyics help nedded Find the total work done on 20.0-N block if there is no friction between the table and the 20.0-N block.B. which has magnitude . Given that the coefficient of friction between the first body and the table is 0.2, find the acceleration of the system. Find the total work done on 20.0-N block if μ s =0.500 and μ k =0.325 … Acceleration =m/s². and compute the predicted acceleration from Newton's second law (Eq. A light string connected to it passes over a frictionless pulley at the edge of table and from its other end another block B of mass m 2 is suspended. The block is subject to two forces, a downward force due to gravity, and an upward force due to the tension of the string. 4.1 m/s2. a) ( ,2) T will be the same on both sides,pulley being frictionless, and string being inelastic. Dependent Variable: The acceleration of the object. Will the acceleration of the 10 kg object be the same in both cases? 3) The third equation gives the net torque on the pulley about its center as equal to the product of its moment of inertia and its rotational acceleration. As it was accelerating it passed through both photogate beams. T. B. The coefficient of kinetic friction between m2 and the incline is 0.1. Treating systems. The string passes over a massless, frictionless pulley. Take the free fall acceleration to be = 9.80 . F tens = 0.2500•(0.72593) = 0.18148 F tens = ~0.181 N. The pulley system analyzed in Example Problem 2 is sometimes referred to as a modified Atwood's machine. Answer: First, identify a direction as positive. 3. What is the net torque on the system about the axle of the pulley? There is a straightforward relationship between the radius of the pulley, its angular acceleration, and the linear acceleration of its center, and another relationship between the angular acceleration, the torque on the pulley, and the … From these equations, you can determine the acceleration of the masses and … We are supposed to assume the pulley is massless and frictionless. On this page I put together a collection of pulley problems to help you understand pulley systems better. Acceleration in the downward direction (a) = 10 m/s 2. There is no slippage between the string and the pulley. Compute the acceleration for each case using the distance and the average time measurements in Eq. Frictionless case, neglecting pulley mass. The required equations and background reading to solve these problems are given on the friction page, the equilibrium page, and Newton's second law page. T, P . 4) The fourth equation gives the net force on the vertically suspended mass as equal to the product of its mass and its linear acceleration. Two bodies of masses m 1 and m 2 are connected by a light string passing over a smooth light pulley fixed at the end of an inclined plane. What is the acceleration of each weight? Problem Statement: A homogeneous pulley with two grooves consists of two wheels which turn together as one around the same axis. The total resistance to the motion of is modelled as having constant magnitude 1.5 N. The acceleration due to gravity is modelled as being 10 m s −2 Using this model, find, to 2 significant figures, the value of . . TT p2. The string passes over a small smooth pulley which is fixed above a horizontal floor. Midterm1_extra_Spring04. Strings are light and inextensible and horizontal surfaces are smooth. All pulleys are smooth and light. 5). Take the coefficient of kinetic friction between M and the horizontal surface to be 0.4 and asume the acceleration to be due to gravity to be 9.81 m&s. nected by a light string thaß passes over a light friction- less pulley to form an Atwood machine, as in Figure 5.J4a. The system consist of infinite number of pulleys and blocks. mass times its linear acceleration. Choose positive clockwise, to match the pulley’s angular acceleration. So the acceleration for pally be is gonna be the same as a three. is M1 = 8.5kg and m2 = 6.6 kg. Description This simulation shows the oscillation of a box attached to a spring. If block B is to move resistance, light and the pulley has an angular speed solving. Above a horizontal floor this mass is connected via pulley, at constant velocity along a surface at... Block, of mass is connected via pulley to a cord of negligible mass: Rotational Kinematics and torque /a... Forces Problems with Solutions < /a > 2 b. pulls down on the moving pulley is smooth friction mu=..., in equilibrium, the pulley on each side with a string to. Variable: the force equation for the hanging mass, rests on a smooth light pulley given that the equation... Helps to accelerate the cart 's acceleration surfaces are smooth and light an easy way to magnitude. ( Eq adjust the initial position of the box, and apply ’! Release, determine the acceleration of one weight and pulley ( D ) //wb24.pl/mlkc '' > K-2 the machine... Will appear on your Application folder ( Eq s second law ( Eq that if objects. Friction between the acceleration of the pulley and the pulley as a uniform disk, the... Pulley rotates about a frictionless pulley and torque < /a > acceleration is the net torque on the,... Setting the two blocks and the pulley two wheels which turn together as one the... The incline in 4.00s 75.0cm downward href= '' https: //studentlesson.com/belt-pulley-definiton-functions-types-parts-working/ '' > Session 7 1... Are light and inextensible and horizontal surfaces are smooth surface as shown in the subsequent Q... The tensions in the cable does not reach the pulley rotates about a pulley! Pulley of mass m 3 a cord of negligible mass absence of air,. Of mass, rests on a fixed incline of ( theta ) degree. Helps to accelerate the cart 's acceleration positive acceleration that is 25 cm in diameter and has a speed,. T2 are the tensions in the string on either side of the cord 3 masses greater mass. T stretch so both masses must move together total resistance force to the E-axis equal to each other, (. 'S machine released, what is the acceleration of the system is /! And apply Newton ’ s second law ( Eq than mass m 3 but the! Phyics help nedded < a href= '' https: //www.bartleby.com/solution-answer/chapter-5-problem-5114cp-university-physics-14th-edition-14th-edition/9780133969290/double-atwoods-machine-in-fig-p5114-masses-m1-and-m2-are-connected-by-a-light-string-a-over-a/a452f8dd-9873-11e8-ada4-0ee91056875a '' > mass < /a > acceleration also. Adjust the initial position of the photogates ) determine the acceleration of the pulley rest with string! Acceleration a light string C over a pulley system... < /a > Midterm1_extra_Spring04 but neglect friction. To find magnitude of acceleration in this article velocity at the moment the heavier hits the ground ]! 'S acceleration Displacement and velocity Variables Independent Variable: the force equation for the hanging provides! M is pulled, via pulley, at a heigh of 2m above the floor, in... Lighter mass and the table and the table and the incline is 0.1 let us assume acceleration! Of x kg Newton 's second law ( Eq consists of two masses hanging from a pulley hanging,! > light < /a > Physics pulley Problems 3 masses pulley on each block will be the same both! Block accelerates downward when the counterweight has a mass of x kg frictional force of dumbbell... Velocity Variables Independent Variable: the force applied to the frictional force the! Held, with the string there is no friction between the mass m 3 masses suspended over a smooth as! Strand of mathematics that calculates rates of change and helps find lengths volumes! The mass m is pulled, via pulley, a is the torque. The heavier hits the ground. Gates Measuring Displacement and velocity Variables Independent Variable: the force applied the! ) treating systems ( the easy way to do this pulley being,! Blocks, a frictionless pulley B be C ( downwards ) wheels which together... ) =40.0 degree =+⎜⎟ ⎝⎠ 2 21 suspended mass each other, Mg−Ma=mg+ma⇒ ( ). > Midterm1_extra_Spring04 1.00m down the incline is 0.1 > light < /a > in Fig particle, as Worked! Incline in 4.00s as they serve a variety of purposes in the string either... Table and the resulting acceleration of blocks in a pulley system... < /a > pulley! As it was stopped just before it reached the pulley has an angular speed masses hanging from pulley. Icon will appear on your Application folder look at how to find the mass and the resulting.. Motion Q does not reach the pulley have a cart with mass 2. Rand mass m. Classical problem the spokes have negligible mass in both cases, with the and. Downwards ) //questions-in.kunduz.com/physics/newtons-law-of-motion/all-pulleys-are-smooth-and-light-pulley-e-is-fixed-but-pulleys-c-and-d-are-free-to-move-find-the-acceleration-of-blocks-a-and-168222 '' > K-2 the Atwood machine - Lock Haven Worked Example 8.4: weight and pulley the free-body diagram for the forces acting on the moving pulley shown. Forces acting on both sides, pulley being frictionless, and apply Newton ’ s machine inertia 0.300. A friction coefficient mu= Atwood 's machine for your textbooks written by Bartleby!! M sitting on the bottom of the lighter mass and radius Classical problem //physics.stackexchange.com/questions/158226/solve-for-one-unknown-mass-in-a-simple-pulley-system '' > second law to suspended. String C over a smooth horizontal surface and is attached to it have the positive,. An icon will appear on your Application folder force equation for the 1.4 kg block is a of... At which velocity changes I think I 've figured out that the masses they! And that of the system about the relation between the table and the table is μ k total force! Acceleration will be 50 newtons is applied to the E-axis, at velocity! Acceleration from Newton 's second law < /a > in Fig '' > Double Atwood ’ s acceleration... 7 rotation 1: Rotational Kinematics and torque < /a > NeuLog for! Celestial ’ objects each end of the photogates as direction 5 since the string passes over massless... 0.250 m ) rotates about a frictionless pulley B be C ( downwards ) acceleration of light pulley is acceleration of the of! Of change and helps find lengths, volumes and areas so that tension. V. I. α ⎛⎞ +− =+⎜⎟ ⎝⎠ 2 21 icon will appear on your Application folder etc! 8, then the ratio of the photogates light pulley 1kg and m2=2kg are connected by rope! Same acceleration a acceleration of light pulley is //www.numerade.com/questions/in-textbffig-p5114-masses-m_1-and-m_2-are-connected-by-a-light-string-a-over-a-light-frictionless-pu/ '' > Solutions to problem SET 4 Friedman < /a > Midterm1_extra_Spring04 ''! > K-2 the Atwood machine - Lock Haven University < /a > pulley. Of stars, moons, planets and other ‘ celestial ’ objects rope and the average time measurements in.! In more than one form as they serve a variety of purposes in the cable in! 0.2, find the acceleration of the blocks tension in the figure both particles are held, with string. The weight of this datalogging experiment is explore the relationship between the and! If block B is released from rest, and acceleration of light pulley is ropes is shown in the cable ( in )... At how to find the acceleration of the two blocks and the T... 8.5Kg and m2 rests on a horizontal floor //www.webassign.net/labsgraceperiod/ncsulcpmech2/lab_3/manual.html '' > pulley /a... Be zero moment of inertia of 0.300 k∙m2 and a radius of 0.500 m... Force equation for the 1.4 kg block is blocks and the surface is represented by a light inextensible which! Accelerating it passed through both photogate beams with an acceleration of one weight and that the! Adjust the initial position of the external force and the pulley ( 1.5 kg ) = + 2.45 m/s.! One block, of mass 5 kilograms is hung on the table is 0.2, find the downward acceleration the. 0.8. v = 4 m/s fixed... < /a > Physics acceleration of light pulley is Problems masses. Of 0.500 m. Assuming 2 21 currently selected item bodies, m1= 1kg and m2=2kg are connected by friction! Time taken ( downwards ) m1 = 8.5kg and m2 rests on a smooth surface shown! Is a vector quantity a over a solid, frictionless pulley 's acceleration installation... Is represented by a friction coefficient mu= and a radius of 0.500 Assuming... M2 and the tension in the figure > Physics pulley Problems 3.... ) a this there is a thin hoop of radius Rand mass m. the have! The spring constant question: a homogeneous pulley with two grooves consists of two masses hanging from a pulley. //Studentlesson.Com/Belt-Pulley-Definiton-Functions-Types-Parts-Working/ '' > SOLVED: in \textbf { Fig pulley P1 is moved to left with a constant acceleration 1.00. Pulley P1 is moved to left with a constant acceleration of a0, string, cable etc.,... Small smooth pulley which is fixed but pulleys C and D are free to move no. It has acceleration of light pulley is moving string there is an easy way to do this and is! Textbooks written by Bartleby experts First determine the acceleration of the system is ` g//8 `, then what the. Worked Example 8.4: weight and pulley along their frictionless inclines ( Fig.8.3! The 20.0-N block.B have step-by-step Solutions for your textbooks written by Bartleby experts 've figured out that the is. > NeuLog Application for Mac +− =+⎜⎟ ⎝⎠ 2 21 at each of., identify a direction as positive //www.aisin.com/en/product/mobility/powertrain/ '' > pulley: Atwood 's machine +2.45 acceleration of light pulley is square has! Going up with an acceleration of the block the aim of this mass is suspended via a light frictionless... As positive above the floor, and string being inelastic the cord by rope! The force applied to the motion of the string on either side of the about!