# Prelab Exercises: Hooke's Law and the Behavior of Springs

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1 59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically from a spring will undergo oscillatory motion if perturbed.. (3 marks) Calculate the effective spring constant of the following arrangement of springs, given that each spring has spring constant k. 3. (4 marks) Draw an arrangement of four springs, each with spring constant k, with an effective spring constant of k.

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3 6 Hooke's Law and the Behavior of Springs Equipment Four 5mm light springs, 5mm tight spring, tapered brass spring, long and short spring support bar with notches, set of masses (0g-000g), digital balance, rubber bands, 30cm ruler, meter stick, laboratory stand with right angle bar clamp, stopwatch. Purpose To use and understand Hooke's law. To examine the behavior of springs and compare the results with Hooke's law. To measure the spring constant of springs using both the static and dynamic methods. To measure the spring constant of springs in parallel and series combinations and compare with theoretical predictions. To examine the behavior of an elastic band and compare the results with Hooke's law. Theory In 678, Robert Hooke investigated the elastic properties of various objects. Hooke measured the extension of the object resulting from the application of a stretching force. The extension of an object is the length that the object is stretched beyond it's resting length. For many objects, including metal wire and coil springs, he observed elastic behavior. Specifically, in a certain range of extension, the material would exert a restoring force opposing the stretching force. In the absence of the stretching force, the material would regain it's original length. He also found that a linear relationship existed between the applied force and the resulting extension in the elastic range. If, for an object, the relationship between applied force and extension is linear, it is said to obey Hooke's law. A spring is a common example of an object that obeys Hooke's Law. Hooke's law for a spring is given by F kx F, () 0 where F is the force exerted by the spring, k is the spring constant, x is the extension of the spring, and Fo is the threshold force of the spring. The threshold force is the minimum force necessary before the spring will begin to stretch. In some springs the coils are pressed together tightly and an initial threshold force is required to separate the coils. In others, the coils are already separated when unextended, so the threshold force is simply zero. Once the coils have separated, the extension is non-zero and Hooke's law applies. There is also a maximum extension above which Hooke's law no longer applies. This limit is known as the elastic limit. Below the elastic limit, the spring deforms elastically, meaning that it will regain it's original shape in the absence of a load. Above the elastic limit, plastic deformation or permanent deformation occurs, Hooke's law is not obeyed, and the original shape of the spring is not regained. In this experiment, we only deal with extensions of springs less than the elastic limit. If a mass, m, is carefully suspended from the free end of a hanging spring so that it is at rest, the spring will stretch to an equilibrium extension, xeq. At this extension, the upward force, F s, exerted by the spring just balances the force of gravity, mg, on the spring. In this situation, illustrated in Figure, we have x mg F kx F. () eq 0 s eq By varying the load on a spring and observing the resulting extension, it is possible to determine the spring constant. We will call this the static method of determining the spring constant. m F s mg Figure If the mass is not at rest, but is suspended from a hanging spring so that there is no horizontal swinging motion, the mass will oscillate in the vertical direction about the equilibrium extension. Suppose the mass is above the equilibrium position, xeq. The force due to gravity is greater than this upward spring force and there will be a downward acceleration. At the equilibrium position, the force of gravity is balanced by the upward spring force, there is no net force and zero acceleration. If the

4 6 mass is below the equilibrium position, then the spring force is greater than the force of gravity and there is an upward acceleration. We can see that there will be a net force acting on the mass which tends to restore it to the equilibrium position. If the spring obeys Hooke's law, the restoring force will be directly proportional to the distance from the equilibrium position. Extension x eq Now, consider a mass that is held at a distance A above the equilibrium position and released. A resulting downward acceleration will be observed. The mass will then overshoot the equilibrium extension and come to rest for an instant at a distance A below the equilibrium position. The mass will accelerate upwards, once again overshooting the equilibrium position to come to rest for an instant at a distance A above the equilibrium position. In a frictionless system, the cycle shown in Figure would continue forever if uninterrupted. This type of motion, in which the acceleration is directly proportional to the distance from the equilibrium position, is called simple harmonic motion. The quantity A, shown in Figure, is the amplitude of the oscillation. The period, T, also shown in Figure, is the time that is required for the mass to undergo one complete cycle. In a real situation, there is internal spring friction as well as resistance due to the surrounding air. With friction, the oscillations are damped, which means the amplitude of the oscillations will decrease with time and the mass will eventually come to a halt at the equilibrium position. In the laboratory setup, the damping is relatively weak and an oscillating mass will only come to rest after a significant period of time. The period of oscillation for a mass hanging from a spring is given by where k is the spring constant of the spring. We can also write T T m, (3) k 4 k m. (4) Equation 4 shows that we can calculate the spring constant by varying the mass and measuring the resulting period of oscillation of a mass suspended from a hanging spring. We will call this the dynamic method of determining the spring constant. It is possible to have multiple spring arrangements and to examine the combined behavior of springs. In particular, we will be investigating two ways of combining springs, a parallel configuration and a series configuration. A parallel configuration of two springs, with spring constants k and k, is shown in Figure 3. We can derive the effective spring constant of this two spring system, meaning that the system of F = k x T x m Figure 3 Figure F = k x two springs can be modeled by a single spring with one spring constant, k'. In this derivation, we will neglect the threshold force of the springs and assume that the springs have the same resting length for simplicity. The total force exerted by the two spring system must be equal to the sum of the forces exerted by each spring. We have F k x k x k x. (5) So, k, the effective spring constant of the system, is given by k k k. (6) Similarly, it can be shown that for a parallel system of n springs, the effective spring constant is given by k k k k 3... k n (7) where k i is the spring constant of the i th spring. A Time

5 63 Figure 4 shows two springs, with spring constants k and k, in a series configuration. For simplicity, the threshold force of the springs and the mass of the springs will be neglected. In the laboratory experiments to be performed here, these are valid approximations. In this case, the total force exerted by the two spring system is not equal to the sum of the forces exerted by each spring. In static equilibrium, the two springs must be exerting the same force, F. From Figure 4, we see that F k x k x. (8) x x m F= k x F= k x In order to find the effective spring constant of the system, we must put the force in terms of the total extension, x. The total extension is given by x x x. (9) We are looking for an expression involving the effective spring constant, k, in the following form From equations 8 and 0, we have Rearranging equation yields, F k x k ( x x ). (0) k ( x x ) k x. () Figure 4 and with equation 8, ( x x ) k k x k k k k x k x, (). (3) Similarly it can be shown that for a system of n springs in series configuration, the effective spring constant is given by where k i is the spring constant of the i th spring.... (4) k k k k k 3 n

6 64 Experimental Procedure. Determine the spring constant of one of the 5mm light springs using the static method. Do this by hanging the spring from the spring support bar and by suspending the weights from the bottom of the spring. Make a table of values of force on the spring and the resulting extension beyond the resting length, together with errors. The mass should remain below 50 grams so that the elastic limit of the spring is not exceeded.. Determine the spring constant of parallel and series arrangements of the 5mm light springs using the static method. It is sufficient to make quick "one data point" measurements of force and extension by suspending a weight on the end of the spring arrangement and measuring a single extension. Do this for arrangements of springs in parallel, 4 springs in parallel, springs in series, and 3 springs in series. Remember to include the mass of the lower spring support bar when calculating the force on the parallel arrangements of springs. 3. Determine how the extension of an elastic band varies with stretching force. In a manner similar to step, make a table of values of force on the band and the resulting extension beyond the resting length. For safety, try not to stretch the elastic band to the point of breakage. 4. Determine the spring constant of one of the 5mm light springs using the dynamic method. Do this by measuring the period of oscillation of various masses suspended on the free end of the spring. The period can be determined by counting the number of oscillations that occur in a length of time measured by the stopwatch. Make a table of values of the period of oscillation and the corresponding mass, with errors. The mass should range from 0 grams to 50 grams. This range is such that the spring can oscillate freely and the elastic limit is not exceeded. 5. Perform any optional investigations as required. Error Analysis Masses measured by the digital balance can be considered to be exact. The stopwatch is subject to errors arising from the reaction time of the person operating the stopwatch. The reaction time can be estimated by double clicking the stopwatch and doubling this to get the error in the time measurement. The error in time measurements can be minimized by timing for a large number of oscillations since the reaction time error is spread over all of the oscillations when propagated to an error in the period. The error in length measurements can be taken to be half of the smallest division of the ruler. There are systematic errors arising from inhomogeneities in the springs (in both coil separation and wire diameter) which cause deviation from ideal behavior. Internal friction in the springs and air resistance contribute to damping which is not accounted for in the theory. Quantities such as the threshold force and the mass of the springs in the analysis of series and parallel configurations of springs have also been neglected.

7 65 The following pages are to be filled out and handed in at the end of this lab session. Hooke's Law and the Behavior of Springs Total Marks = 7 Name: I.D. #: Lab Section: Lab Instructor:. ( marks) Enter numerical values for mass, m, force, F, length, L, and extension, x, of the 5mm light spring as measured in procedure step, together with errors. Proper headings and units should be included in the table. Unextended length of the spring:. ( marks) Plot a graph of F versus x for the 5mm light spring. If the behavior of the spring is described by Equation, then the graph should be linear. Include errors bars on the graph as well as proper axis labeling and a title for the graph. 3. ( mark) From the slope and intercept of the graph, determine the spring constant as well as the threshold force of the spring. Does Hooke's law apply to the spring? Why or why not?

8 66 4. (3 marks) Fill in the blanks with your measurements from procedure step. When calculating the effective spring constants, you must neglect the threshold friction of the system. Assuming that the springs are identical, equations 7 and 4 can be used to predict the effective spring constant using the results of the previous exercise. springs in parallel Data and calculations: 4 springs in parallel Data and calculations: Measured effective spring constant: Predicted effective spring constant: springs in series Data and calculations: Measured effective spring constant: Predicted effective spring constant: 3 springs in series Data and calculations: Measured effective spring constant: Predicted effective spring constant: Measured effective spring constant: Predicted effective spring constant: 5. ( marks) Enter numerical values for mass, m, force, F, length, L, and extension, x, of the elastic band as measured in procedure step 3, together with errors. Proper headings and units should be included in the table.

9 67 Unextended length of the elastic band: 6. ( marks) Plot a graph of F versus x for the elastic band. Compare this graph qualitatively with that of the spring. Does Hooke's law apply to the elastic band? Why or why not?

10 68 7. ( marks) Enter numerical values for mass, m, number of oscillations, N, total time, t, period, T, and period, T, for the 5mm light spring as measured in procedure step 4, together with errors. Proper headings and units should be included in the table. 8. ( marks) Plot a graph of T versus m. If the behavior of the spring is described by Equation 4, then the graph will be linear. Include errors bars on the graph as well as proper axis labeling and a title for the graph. 9. ( marks) From the slope of the graph, determine the spring constant of the spring. Compare the value of the spring constant with that obtained using the static method. 0. ( bonus mark) Determine the threshold force of the 5mm tight spring. Do this by making a data table of force on the spring and the resulting extension in a range around the threshold force (some of the data

11 69 points should be below the threshold force). Enter numerical values for mass, m, force, F, length, L, and extension, x, of the small stiff spring, together with errors. Proper headings and units should be included in the table. Plot a graph of F versus x and extrapolate to find the threshold force of the spring. Unextended length of the spring: Threshold force of the spring:. ( bonus mark) In the analysis leading to Equation 4, the mass of the spring was neglected. We can account for the mass of the spring by modifying Equation 4 to include the effective mass of the system. There is kinetic energy associated with a mass which is oscillating. We wish to find the kinetic energy associated with the oscillating spring. Due to the nature of the spring, each point along the length of an extending spring is moving with a different velocity. A spring, with mass m s, is extending with end point velocity v, as shown in Figure 5. The kinetic energy of this spring is given by E k ( s p r in g ) m s 3 v. (5) m s Since v is the velocity of the hanging mass, the kinetic energy associated with the hanging mass is simply E k ( m a s s ) mv. (6) m v So then the total kinetic energy of the expanding spring system is Figure 5 m E E E m k ( to ta l ) k ( s p r in g ) k ( m a s s ) 3 s v (7) and we can see that the effective mass of the system is m. 3 In order to take into account the mass of the spring in the analysis of the oscillatory motion of the hanging mass, we simply need to re-derive Equation 4 with the corrected effective mass of the system. The result is then m s

12 70 T 4 k m m s 3. (8) We can see that, given a significantly massive spring, the mass of the spring can be calculated from the analysis of it's oscillatory motion. Determine the spring constant and the mass of the tapered brass spring using the dynamic method. Make a table of values of the period of oscillation and the corresponding mass, with errors. The mass should range from 50 grams to 50 grams so that the spring can oscillate freely and the elastic limit is not exceeded. Make sure to weigh the tapered brass spring using the digital balance so that you can compare your results. Enter values for mass, m, number of oscillations, N, total time, t, period, T, and period, T, for the heavy spring, together with errors. Headings and units should be included in the table. Mass of the spring:. ( bonus mark) Plot a graph of T versus m for the tapered brass spring. Include errors bars on the graph as well as proper axis labeling and a title for the graph. From the slope and intercept of the graph, determine the spring constant as well as the mass of the spring using equation 8. Compare the calculated value of the mass of the brass spring with the one that you measured.

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