PART B. DYNAMIC METHOD In this part, the spring constant & is measured using a dynamic load (a) Start with a AM of just the mass hanger. Set the mass oscillating without causing the spring to become fully compressed (these need only be small oscillations). Measure the time for the mass to complete 20 oscillations. Divide the time by the number of oscillations to determine the period ? and then compute ?' (b) Repeat step (a) adding mass to the mass hanger in 50 g increments Table 2: Determining the spring constant k using an oscillating load AM (KR) for 20 cycles (s) 0 050 me banger sand 0.100 10.150 0.200 0.250 0.300 0.350 10 400 (c) Make a graph of your data by plotting ?" in s on the vertical axis and AM in kilograms on the horizontal. Determine the slope of your graph and, from this, the spring constant &. How does the value you get for & from the dynamic method compare with your value you found using the static method?Oscillations (vibrations) and waves are intimately related subjects. All waves are created by vibrating sources. This lab introduces the basics of oscillatory motion including Hooke's law. simple harmonic motion, and resonance. In a later lab you will explore resonance phenomena involving sound waves. Before the Lab: Prepare the usual "Summary of purpose and method" and "Theory" sections. Theory: Many systems in nature undergo motion that is periodic in time and thus can be characterized by a parameter 7, called the period; this is simply the time required for one cycle of the motion. For instance, the earth revolves around its axis on a regular basis and has a period 7 of 24 hours. In many periodic systems the motion is an oscillation back and forth about some central equilibrium position in which case an important quantity becomes the distance-used here in a very loose sense-that the system has been displaced from equilibrium. The simple pendulum you used in the first lab is one example of such a system. The pendulum naturally comes to rest at the bottom of the are but, if pulled back from this equilibrium position, the pendulum will oscillate back and forth. This motion can be characterized by a function that gives the pendulum's displacement from the center position at any time . For many systems the displacement function assumes the particularly simple form x(1) - Acos(271/T)- Acos(27 / 1) where the equilibrium position is x - 0, frepresents the frequency, which indicates how quickly the system is oscillating (specifically, the system oscillates / times every second) and of is the amplitude of the oscillation (the maximum distance from the equilibrium position. This is the formula for simple harmonic motion (S.H.M.), and it forms the basis for the description of an amazingly wide variety of physical phenomena. The system you will be examining is a mass suspended from the end of a spring. In Part A you will look at the system at rest to determine the key parameter of the spring while in Part B you will determine the same parameter by examining the system in motion. In Part C you will look at transfer of energy to the spring system. If the mass is not in motion then Newton's first law tells us that the vector sum of all forces acting on the mass must be zero. Thus, when the mass is at rest the force of the spring, which acts upward. must cancel with the downward force due to gravity (i.e. F. - -mg, where we are taking down as the positive direction). As first proposed by Robert Hooke, a contemporary and rival of Newton, the magnitude of the spring force is, to a good approximation, proportional to the stretch of the spring from its unstretched position (called Ax): F. - -KAX, (2) 31PART C. ENERGY TRANSFER IN OSCILLATORY SYSTEMS This part will be done as a group with the TA. Record observations and discuss the results. As you are aware, resonance occurs when a periodic force (often called a driving force) is applied to an oscillating system at the same frequency at which the system naturally oscillates, In this part you will use two springs attached to a common rod, each with suspended masses, and consider the efficiency of energy transfer from one spring to the other. (n) Hang the thick black spring (hereafter referred to as the black spring - the spring used in parts A and B will be referred to as the gold spring) in the notch nearest the clamp and suspend 900 g on the weight pan. Get a rough value of the period by counting 20 periods and dividing by 20, as before. for 20 oscillations (b) You should be able to depress the end of the rod slightly just by pushing down on it. Set the spring at rest and try to get the spring to oscillate by pushing gently on the end of the rod at the spring system's natural frequency. If you are successful, the amplitude of the spring's oscillation will greatly exceed the amplitude of the oscillation you are creating in the rod Can you induce significant oscillations in the spring by pushing at any frequency other than the natural frequency? (c) Now, hang the gold spring in the notch nearest the end of the rod. Hang an empty weight pan on the end of this spring. (d) Stretch and release the black spring. Allow this system to oscillate for a minute or more, and comment on the amount of energy transferred to the gold spring. Repeat several times, each time adding a 50 g mass to the gold spring Table 3: Transfer of energy from loaded black spring system to gold spring system AM (Kg) amount and pattern of energy transfer 0.050 (mass hanger) 0.100 0.150 0.200 (e) What comments can you make on the conditions for the gold spring to oscillate? (Recall that you have recorded the period for the natural frequency of the gold spring for a variety of loads in Table 2.)Procedure: PART A. STATIC METHOD his part, the spring constant & is measured using a static load (a) Secure the upper end (the wider end) of the less-stiff, gold coloured spring to the support and attach a weight hanger to the lower end of the spring. Put sufficient mass on the pan that the coils of the spring are completely separated (50 - 100g). The mass at this point will be called M, (include the mass of the hanger). Use a set square and the scale on the vertical support to measure the position t. of the pan. (b) Add an additional mass AA - 50 g to the pan and record the new value of r. Continue to add mass in 50 g increments. Record your data in Table 1. You can assume that the uncertainty on the mass is negligible. M. - Table 1. Determining the spring constant using a static load AM (kg) x (m) Ar = x x. (m). 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 (d) Make a graph of your data by plotting x (in meters) on the vertical axis and AM/ (in kilograms) on the horizontal. From the slope of this graph, estimate a value for the spring constant & and estimate the uncertainty in the value for &.In Pan A will suspend a series of masses from a spring and measure the resu placement. Since both the mass of the spring itself and the mass of the weight pan apply force at tend to stretch the spring, we can write the mass as m - M. . AM, where M, is the fixed initial mass stretching the spring and AM is the additional amount that has been added. You will make measurements for a number of different values of AM. We write the displacement as s " r. + At where r is the current scale reading.. x. would have been the scale reading when no downward force was applied (be. the unstretched position of the spring) and Ar is the displacement from the unstretched position. Show that using these conventions and F. " - mg, equation (2) can be written as (3) A graph of x versus Alf should yield a straight line. By comparing equation (3) with the standard form for a straight line, y = me + b, determine what the slope of the graph would be in terms of the variables given. Rearrange this expression to show how & can be calculated from the slope. If the suspended mass is displaced from equilibrium (i.c. pulled down slightly and released) then the mass will undergo S.H.M. according to equation (1). As shown in your text, the frequency, f, which is equal to 1/T, is related to the mass and the spring constant by (4) Here, we use the notation f. to indicate that this is the system's natural frequency. It is possible to make the system oscillate at other frequencies, but this is the frequency at which it oscillates if you release it and leave it alone. Show that if m = M. + Alf this can be rewritten as - AM - AM. (5) Determine the expected slope if 7" is plotted versus AM. Rearrange this expression to show how & can be calculated from the slope. " C of the lab you will investigate transfer of energy to the system and resonance. When a odie force is applied to a harmonic system, the system can be made to oscillate with the same as the force. However, a resonant frequency usually exists (there may be more than one. depending on the system) where the amplitude of the oscillations becomes very large (maximum zy has been transferred to the system). The value of f resulting in resonance is given by (6) licating that the re is the same frequency ystem would oscillate by