Figure3.11 Two objects linked together Figure 3.12

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1 Figure3. Two objects linked together Another common situation involves two or more objects connected together by a cable or rope, such as a van pulling a rope connected to a wagon that is pulling a rope connected to a second wagon. Perhaps the first scientific quantitative application of Newton s second law to a situation with two objects connected together by a string or rope involved two blocks of different mass at the ends of a string that passed over a pulley (Fig. 3.). George Atwood invented this experiment and used it to determine the value of g. In 784 he published a description of the apparatus, which became affectionately known as the Atwood machine. Atwood was one of the early scientists who described experimental results mathematically. He was famous for his lectures, particularly his lecture demonstrations. William Pitt, an English prime minister who attended Atwood s lectures, enlisted him to help apply mathematical models to economic issues in England. How did Atwood use it to determine free-fall acceleration g? Figure 3. Atwood Machine Remember, this was 784. There were no motion sensors, no precision stopwatches; nothing that would allow the accurate measurement of the motion of a rapidly accelerating object ( 9.8 m s would have been considered a rapid acceleration in those days). Atwood s machine was significant in that it allowed the determination of g despite these difficulties. Observations of Atwood s machine show that when the blocks are of different mass, and the system is released from rest, the heavier block always accelerates downward, and the lighter block always accelerates upward. For the blocks in Fig. 3., no matter how different their masses are, both always have accelerations of the same magnitude (since they are attached to one another by a string), and the magnitude of this acceleration was always less than g. If the blocks are nearly the same mass, the magnitude of their acceleration is small and measurements of the time interval it took them to Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-3

2 move a certain distance are made much more easily. By using Newton s nd law, Atwood devised a way to relate these time measurements to g. Effect of Pulleys in Two Object Problems Before we analyze an Atwood type apparatus, we need to consider the effect of the pulley on such a system. Consider the situation shown in Fig. 3.3a. The string on the left side of the pulley exerts a force T! S on of magnitude T on block and on the right side a force T! S on of magnitude T on block. How does T compare to T? To answer this question, it is easier to consider a different device that changes the direction of the string but does not rotate (Fig. 3.3b). If the string has a small mass compared to the mass of each block, and there is no friction between the string and the surface of the smooth circular curved surface over which it passes, then the forces exerted by the string on each side of that surface are equal ( T " T ). Figure 3.3 Effect that pulley has on string force on each side One way to think about this is to consider the string as the system. Block on the left side pulls down on the string with a force of magnitude T and block on the other side pulls down on the string with a force of magnitude T. The curved surface pushes up on the string. If the forces that the blocks exert on the string are not equal in magnitude (for example if T + T), then the string of mass m slides across the curved surface with an acceleration of magnitude a T % T m ". The mass of the string is typically very small and if T + T, the string would have a very large acceleration. In an Atwood machine, the blocks never have acceleration greater in magnitude than g and the string moves with the same acceleration as the blocks. This means that T " T; the curved surface simply changes the direction of the string. If the string passes over a pulley instead of a smooth fixed curved surface, the string does not slide but instead causes the pulley to turn. If T " T, the axle of the pulley needs to be frictionless so that it does not impede the turning pulley. In addition, the mass of the pulley should be very small compared to the mass of each block. If the pulley does have a very low mass, and there is no friction impeding its motion, then using a similar argument to the one in the last Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-4

3 paragraph (but with the pulley as the system of interest) unequal string forces would cause a very large turning acceleration of the pulley. Again, since the string can t accelerate with a magnitude larger than g, the forces that the strings exert on the blocks on each side of the pulley must have equal magnitudes ( T " T ). Summary about Pulleys Low mass pulleys that rotate without friction in the bearing change the direction of a low mass string passing around the pulley but do not change the magnitude of the force that the string exerts on objects attached to either of its ends. Atwood type machines We can now analyze a situation that involves a string and pulley. An example is a modified Atwood machine one block on a smooth horizontal surface being pulled by a string passing over a pulley to a second hanging block. Let s see how Atwood would have analyzed this situation. Example 3.7 Is a fall always a free fall? A modified Atwood machine is shown in Fig. 3.4a. Determine expressions for the force that the string exerts on the hanging block of mass m and the magnitude of the acceleration of each block when they are moving freely. Figure 3.4(a) Modified Atwood machine Sketch and Translate The two blocks have masses m and m. If nobody is holding block, block moves down and pulls the string attached to block. Both blocks move with the same increasing speed block on the horizontal surface and block downward. The hanging block has a downward acceleration and the sliding block has the same magnitude acceleration toward the right. Our goal is to find the force exerted by the string on block and the magnitude of the accelerations of the blocks. Simplify and Diagram Assume that the pulley s mass is very small and rotates without friction. Assume also that there is no friction between the horizontal surface and block. Force diagrams for each block and our choice of coordinate systems are shown (Fig. 3.4b). For block, Earth exerts a downward gravitational force F! E on, the table surface exerts an equal magnitude upward normal force on it N! T on, and the string exerts an unknown horizontal force toward the right on block T! S on. For block, Earth exerts a downward gravitational force Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-5

4 on F! E on, while the string exerts an unknown upward force on T! S on. As noted earlier, the string exerts the same magnitude force on block as it does on block ( T S on " T S on " T ). Figure 3.4(b) Represent Mathematically Now, use the force diagrams to help apply Newton s second law in component form for each block. For block, the forces in the vertical y direction balance since the vertical component of its acceleration is zero (for this problem we will not even write the y - component equation as it is not needed). The x -component form of Newton s second law is: ma " ' F x x Note that by inspection of the force diagram, the only non-zero x-component force is the force exerted by the string on block : m " T " T a x S on x For block we only need to look at the vertical direction, as there are no forces exerted on it in the horizontal direction. Notice that we chose the vertical y -axis pointing downward so that the y -component of block s acceleration equals the x -component of block s acceleration. The y -component form of Newton s second law for block is: ma " ' F y y ma " T # F y S on y E on y Noting that TS on y " T and FE on y ", we get: ma " T# y As noted earlier, the blocks have accelerations of the same magnitude, to the right and down. Thus, a, x " a, y " a. Solve and Evaluate After substituting a for the accelerations in ma x y " T and ma " T#, we have two equations with the same two unknowns ( T and a). ma " T Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-6

5 ma" T# After substituting the expression for T from the first equation into the second, we get: ma" ma#. Moving the terms containing the acceleration to the left side, factoring out a, and dividing both sides by the sum of the masses, we have: a " m # m The acceleration is less than g! We can now find the force exerted by the string on the hanging block by inserting the above expression for the acceleration into ma = T:, m - T " ma " m ". /. m# m 0m# m Are the results reasonable? Consider the equation for the acceleration a " m # m. The only downward pointing force exerted on the hanging block is the gravitational force. But, because the two blocks are connected together, this force has to cause the sum of the masses ( m # m ) to accelerate. We can also use limiting case analysis to check the expressions for the acceleration and the force that the rope exerts on the blocks. Suppose, for example, that block has zero mass. Then block should fall with free fall acceleration, and the string should not exert a tension force on either block. This is what the two equations predict: a 0 # m " " g m( g and T " m " 0( " 0. m # m 0 # m Try It Yourself: Determine an expression for acceleration for a real Atwood machine for which two objects of mass m and m move in the vertical direction (m > m ). Answer: ( m% m) g a " m # m We see now how the Atwood machine can be used to determine the acceleration g. You need two objects of almost equal mass and significantly heavier than the string and the pulley, and a low friction pulley. With this setup, the two objects will have a very small acceleration. If you measure the distance one of them travels ( * y ) during a particular time interval ( * t ) after being released from rest, you can use the expression * y " a *t (this is an application of one of the equations of kinematics) to determine the acceleration of the objects. Using this value of Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-7

6 acceleration and the values for their masses, you can use the expression you determined in the previous Try It Yourself to determine the value of g. Review Question 3.3 For problems involving objects moving upward or downward along inclined surfaces, we choose the x -axis parallel to the surface, and the y -axis perpendicular to the surface. Why not use horizontal and vertical axes? 3.4 Friction In previous sections, we assumed that objects moved across absolutely smooth surfaces no friction. The vast majority of processes encountered in the real world involve some degree of friction. In this section we will construct mathematical models of friction so that we can take friction into account quantitatively. Static friction Consider some simple experiments that involve pulling a block with a spring scale, as shown in Observational Experiment Table 3.. The spring scale exerts an increasing force on the block. Observe carefully what happens to the block. ALG Observational Experiment Table 3. Pulling a block with a spring scale Observational experiments Analysis A block is at rest on the horizontal surface of a desk. A spring scale pulls lightly on the block that is at rest on a horizontal surface; the block does not move. The spring scale pulls harder on the block at rest on the horizontal surface; the block still does not move. Etkina/Gentile/Van Heuvelen Process Physics /e, Chapter 3 3-8